LMFDB - Embedded newform 8012.2.a.b.1.6

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:	$0$
Dimension:	$88$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.80891 q^{3} -0.455894 q^{5} -2.00885 q^{7} +4.88996 q^{9} +O(q^{10})$	$q-2.80891 q^{3} -0.455894 q^{5} -2.00885 q^{7} +4.88996 q^{9} -4.59808 q^{11} -0.592441 q^{13} +1.28056 q^{15} -6.80638 q^{17} +5.98468 q^{19} +5.64267 q^{21} +5.56126 q^{23} -4.79216 q^{25} -5.30872 q^{27} -1.71028 q^{29} -0.170513 q^{31} +12.9156 q^{33} +0.915822 q^{35} +2.97472 q^{37} +1.66411 q^{39} -11.1841 q^{41} -5.44843 q^{43} -2.22930 q^{45} -4.04656 q^{47} -2.96452 q^{49} +19.1185 q^{51} -7.84089 q^{53} +2.09624 q^{55} -16.8104 q^{57} -7.88757 q^{59} -2.40031 q^{61} -9.82319 q^{63} +0.270090 q^{65} +3.31621 q^{67} -15.6211 q^{69} -7.41409 q^{71} +4.92664 q^{73} +13.4607 q^{75} +9.23686 q^{77} -7.94220 q^{79} +0.241822 q^{81} -7.67098 q^{83} +3.10299 q^{85} +4.80402 q^{87} +2.44487 q^{89} +1.19012 q^{91} +0.478954 q^{93} -2.72838 q^{95} +6.39378 q^{97} -22.4844 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) $ 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9	$ 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9} + 16 q^{11} - 3 q^{13} + 21 q^{15} + 32 q^{17} + 49 q^{19} + 7 q^{21} + 24 q^{23} + 114 q^{25} + 82 q^{27} + 2 q^{29} + 40 q^{31} + 31 q^{33} + 26 q^{35} + 37 q^{39} + 22 q^{41} + 98 q^{43} + 12 q^{45} + 48 q^{47} + 132 q^{49} + 55 q^{51} - q^{53} + 116 q^{55} + 62 q^{57} + 34 q^{59} + 132 q^{63} + 39 q^{65} + 75 q^{67} + 15 q^{69} + 24 q^{71} + 104 q^{73} + 87 q^{75} + 4 q^{77} + 111 q^{79} + 128 q^{81} + 64 q^{83} + 7 q^{85} + 115 q^{87} + 14 q^{89} + 73 q^{91} - 7 q^{93} + 51 q^{95} + 117 q^{97} + 72 q^{99}+O(q^{100}) $ 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9 + 16 * q^11 - 3 * q^13 + 21 * q^15 + 32 * q^17 + 49 * q^19 + 7 * q^21 + 24 * q^23 + 114 * q^25 + 82 * q^27 + 2 * q^29 + 40 * q^31 + 31 * q^33 + 26 * q^35 + 37 * q^39 + 22 * q^41 + 98 * q^43 + 12 * q^45 + 48 * q^47 + 132 * q^49 + 55 * q^51 - q^53 + 116 * q^55 + 62 * q^57 + 34 * q^59 + 132 * q^63 + 39 * q^65 + 75 * q^67 + 15 * q^69 + 24 * q^71 + 104 * q^73 + 87 * q^75 + 4 * q^77 + 111 * q^79 + 128 * q^81 + 64 * q^83 + 7 * q^85 + 115 * q^87 + 14 * q^89 + 73 * q^91 - 7 * q^93 + 51 * q^95 + 117 * q^97 + 72 * 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
$3$	−2.80891	−1.62172	−0.810862	−	0.585238i	$-0.801001\pi$
$3$	−2.80891	−1.62172	−0.810862	+	0.585238i	$0.801001\pi$
$4$	0	0
$5$	−0.455894	−0.203882	−0.101941	−	0.994790i	$-0.532505\pi$
$5$	−0.455894	−0.203882	−0.101941	+	0.994790i	$0.532505\pi$
$6$	0	0
$7$	−2.00885	−0.759274	−0.379637	−	0.925136i	$-0.623951\pi$
$7$	−2.00885	−0.759274	−0.379637	+	0.925136i	$0.623951\pi$
$8$	0	0
$9$	4.88996	1.62999
$10$	0	0
$11$	−4.59808	−1.38637	−0.693187	−	0.720758i	$-0.743794\pi$
$11$	−4.59808	−1.38637	−0.693187	+	0.720758i	$0.743794\pi$
$12$	0	0
$13$	−0.592441	−0.164313	−0.0821567	−	0.996619i	$-0.526181\pi$
$13$	−0.592441	−0.164313	−0.0821567	+	0.996619i	$0.526181\pi$
$14$	0	0
$15$	1.28056	0.330640
$16$	0	0
$17$	−6.80638	−1.65079	−0.825395	−	0.564555i	$-0.809048\pi$
$17$	−6.80638	−1.65079	−0.825395	+	0.564555i	$0.809048\pi$
$18$	0	0
$19$	5.98468	1.37298	0.686490	−	0.727140i	$-0.259151\pi$
$19$	5.98468	1.37298	0.686490	+	0.727140i	$0.259151\pi$
$20$	0	0
$21$	5.64267	1.23133
$22$	0	0
$23$	5.56126	1.15960	0.579801	−	0.814758i	$-0.303130\pi$
$23$	5.56126	1.15960	0.579801	+	0.814758i	$0.303130\pi$
$24$	0	0
$25$	−4.79216	−0.958432
$26$	0	0
$27$	−5.30872	−1.02166
$28$	0	0
$29$	−1.71028	−0.317591	−0.158796	−	0.987311i	$-0.550761\pi$
$29$	−1.71028	−0.317591	−0.158796	+	0.987311i	$0.550761\pi$
$30$	0	0
$31$	−0.170513	−0.0306250	−0.0153125	−	0.999883i	$-0.504874\pi$
$31$	−0.170513	−0.0306250	−0.0153125	+	0.999883i	$0.504874\pi$
$32$	0	0
$33$	12.9156	2.24832
$34$	0	0
$35$	0.915822	0.154802
$36$	0	0
$37$	2.97472	0.489042	0.244521	−	0.969644i	$-0.421369\pi$
$37$	2.97472	0.489042	0.244521	+	0.969644i	$0.421369\pi$
$38$	0	0
$39$	1.66411	0.266471
$40$	0	0
$41$	−11.1841	−1.74667	−0.873333	−	0.487123i	$-0.838046\pi$
$41$	−11.1841	−1.74667	−0.873333	+	0.487123i	$0.838046\pi$
$42$	0	0
$43$	−5.44843	−0.830877	−0.415439	−	0.909621i	$-0.636372\pi$
$43$	−5.44843	−0.830877	−0.415439	+	0.909621i	$0.636372\pi$
$44$	0	0
$45$	−2.22930	−0.332325
$46$	0	0
$47$	−4.04656	−0.590251	−0.295126	−	0.955458i	$-0.595361\pi$
$47$	−4.04656	−0.590251	−0.295126	+	0.955458i	$0.595361\pi$
$48$	0	0
$49$	−2.96452	−0.423503
$50$	0	0
$51$	19.1185	2.67713
$52$	0	0
$53$	−7.84089	−1.07703	−0.538515	−	0.842616i	$-0.681014\pi$
$53$	−7.84089	−1.07703	−0.538515	+	0.842616i	$0.681014\pi$
$54$	0	0
$55$	2.09624	0.282657
$56$	0	0
$57$	−16.8104	−2.22659
$58$	0	0
$59$	−7.88757	−1.02687	−0.513437	−	0.858127i	$-0.671628\pi$
$59$	−7.88757	−1.02687	−0.513437	+	0.858127i	$0.671628\pi$
$60$	0	0
$61$	−2.40031	−0.307328	−0.153664	−	0.988123i	$-0.549107\pi$
$61$	−2.40031	−0.307328	−0.153664	+	0.988123i	$0.549107\pi$
$62$	0	0
$63$	−9.82319	−1.23761
$64$	0	0
$65$	0.270090	0.0335005
$66$	0	0
$67$	3.31621	0.405139	0.202569	−	0.979268i	$-0.435071\pi$
$67$	3.31621	0.405139	0.202569	+	0.979268i	$0.435071\pi$
$68$	0	0
$69$	−15.6211	−1.88055
$70$	0	0
$71$	−7.41409	−0.879890	−0.439945	−	0.898025i	$-0.645002\pi$
$71$	−7.41409	−0.879890	−0.439945	+	0.898025i	$0.645002\pi$
$72$	0	0
$73$	4.92664	0.576619	0.288310	−	0.957537i	$-0.406907\pi$
$73$	4.92664	0.576619	0.288310	+	0.957537i	$0.406907\pi$
$74$	0	0
$75$	13.4607	1.55431
$76$	0	0
$77$	9.23686	1.05264
$78$	0	0
$79$	−7.94220	−0.893568	−0.446784	−	0.894642i	$-0.647431\pi$
$79$	−7.94220	−0.893568	−0.446784	+	0.894642i	$0.647431\pi$
$80$	0	0
$81$	0.241822	0.0268691
$82$	0	0
$83$	−7.67098	−0.842000	−0.421000	−	0.907061i	$-0.638321\pi$
$83$	−7.67098	−0.842000	−0.421000	+	0.907061i	$0.638321\pi$
$84$	0	0
$85$	3.10299	0.336566
$86$	0	0
$87$	4.80402	0.515045
$88$	0	0
$89$	2.44487	0.259155	0.129578	−	0.991569i	$-0.458638\pi$
$89$	2.44487	0.259155	0.129578	+	0.991569i	$0.458638\pi$
$90$	0	0
$91$	1.19012	0.124759
$92$	0	0
$93$	0.478954	0.0496652
$94$	0	0
$95$	−2.72838	−0.279926
$96$	0	0
$97$	6.39378	0.649190	0.324595	−	0.945853i	$-0.394772\pi$
$97$	6.39378	0.649190	0.324595	+	0.945853i	$0.394772\pi$
$98$	0	0
$99$	−22.4844	−2.25977
$100$	0	0
$101$	−2.34435	−0.233271	−0.116636	−	0.993175i	$-0.537211\pi$
$101$	−2.34435	−0.233271	−0.116636	+	0.993175i	$0.537211\pi$
$102$	0	0
$103$	5.32678	0.524864	0.262432	−	0.964951i	$-0.415476\pi$
$103$	5.32678	0.524864	0.262432	+	0.964951i	$0.415476\pi$
$104$	0	0
$105$	−2.57246	−0.251046
$106$	0	0
$107$	−3.85176	−0.372364	−0.186182	−	0.982515i	$-0.559611\pi$
$107$	−3.85176	−0.372364	−0.186182	+	0.982515i	$0.559611\pi$
$108$	0	0
$109$	−10.0714	−0.964666	−0.482333	−	0.875988i	$-0.660211\pi$
$109$	−10.0714	−0.964666	−0.482333	+	0.875988i	$0.660211\pi$
$110$	0	0
$111$	−8.35572	−0.793090
$112$	0	0
$113$	1.05655	0.0993922	0.0496961	−	0.998764i	$-0.484175\pi$
$113$	1.05655	0.0993922	0.0496961	+	0.998764i	$0.484175\pi$
$114$	0	0
$115$	−2.53534	−0.236422
$116$	0	0
$117$	−2.89701	−0.267829
$118$	0	0
$119$	13.6730	1.25340
$120$	0	0
$121$	10.1424	0.922033
$122$	0	0
$123$	31.4152	2.83261
$124$	0	0
$125$	4.46418	0.399289
$126$	0	0
$127$	11.5122	1.02154	0.510770	−	0.859717i	$-0.329360\pi$
$127$	11.5122	1.02154	0.510770	+	0.859717i	$0.329360\pi$
$128$	0	0
$129$	15.3041	1.34745
$130$	0	0
$131$	−5.19808	−0.454159	−0.227079	−	0.973876i	$-0.572918\pi$
$131$	−5.19808	−0.454159	−0.227079	+	0.973876i	$0.572918\pi$
$132$	0	0
$133$	−12.0223	−1.04247
$134$	0	0
$135$	2.42021	0.208299
$136$	0	0
$137$	−6.57848	−0.562038	−0.281019	−	0.959702i	$-0.590672\pi$
$137$	−6.57848	−0.562038	−0.281019	+	0.959702i	$0.590672\pi$
$138$	0	0
$139$	5.56978	0.472423	0.236211	−	0.971702i	$-0.424094\pi$
$139$	5.56978	0.472423	0.236211	+	0.971702i	$0.424094\pi$
$140$	0	0
$141$	11.3664	0.957224
$142$	0	0
$143$	2.72409	0.227800
$144$	0	0
$145$	0.779706	0.0647510
$146$	0	0
$147$	8.32706	0.686805
$148$	0	0
$149$	−11.7392	−0.961710	−0.480855	−	0.876800i	$-0.659674\pi$
$149$	−11.7392	−0.961710	−0.480855	+	0.876800i	$0.659674\pi$
$150$	0	0
$151$	−1.39796	−0.113765	−0.0568823	−	0.998381i	$-0.518116\pi$
$151$	−1.39796	−0.113765	−0.0568823	+	0.998381i	$0.518116\pi$
$152$	0	0
$153$	−33.2829	−2.69077
$154$	0	0
$155$	0.0777356	0.00624387
$156$	0	0
$157$	−15.4493	−1.23299	−0.616493	−	0.787360i	$-0.711447\pi$
$157$	−15.4493	−1.23299	−0.616493	+	0.787360i	$0.711447\pi$
$158$	0	0
$159$	22.0243	1.74664
$160$	0	0
$161$	−11.1717	−0.880456
$162$	0	0
$163$	−9.66812	−0.757265	−0.378633	−	0.925547i	$-0.623606\pi$
$163$	−9.66812	−0.757265	−0.378633	+	0.925547i	$0.623606\pi$
$164$	0	0
$165$	−5.88813	−0.458391
$166$	0	0
$167$	4.62793	0.358120	0.179060	−	0.983838i	$-0.442694\pi$
$167$	4.62793	0.358120	0.179060	+	0.983838i	$0.442694\pi$
$168$	0	0
$169$	−12.6490	−0.973001
$170$	0	0
$171$	29.2648	2.23794
$172$	0	0
$173$	−19.7306	−1.50009	−0.750046	−	0.661385i	$-0.769969\pi$
$173$	−19.7306	−1.50009	−0.750046	+	0.661385i	$0.769969\pi$
$174$	0	0
$175$	9.62673	0.727713
$176$	0	0
$177$	22.1554	1.66531
$178$	0	0
$179$	−11.7420	−0.877637	−0.438819	−	0.898576i	$-0.644603\pi$
$179$	−11.7420	−0.877637	−0.438819	+	0.898576i	$0.644603\pi$
$180$	0	0
$181$	8.49080	0.631116	0.315558	−	0.948906i	$-0.397808\pi$
$181$	8.49080	0.631116	0.315558	+	0.948906i	$0.397808\pi$
$182$	0	0
$183$	6.74225	0.498401
$184$	0	0
$185$	−1.35616	−0.0997067
$186$	0	0
$187$	31.2963	2.28861
$188$	0	0
$189$	10.6644	0.775723
$190$	0	0
$191$	22.9982	1.66409	0.832045	−	0.554708i	$-0.187170\pi$
$191$	22.9982	1.66409	0.832045	+	0.554708i	$0.187170\pi$
$192$	0	0
$193$	−15.0475	−1.08314	−0.541570	−	0.840655i	$-0.682170\pi$
$193$	−15.0475	−1.08314	−0.541570	+	0.840655i	$0.682170\pi$
$194$	0	0
$195$	−0.758658	−0.0543286
$196$	0	0
$197$	−8.50499	−0.605956	−0.302978	−	0.952998i	$-0.597981\pi$
$197$	−8.50499	−0.605956	−0.302978	+	0.952998i	$0.597981\pi$
$198$	0	0
$199$	−7.83493	−0.555404	−0.277702	−	0.960667i	$-0.589573\pi$
$199$	−7.83493	−0.555404	−0.277702	+	0.960667i	$0.589573\pi$
$200$	0	0
$201$	−9.31491	−0.657023
$202$	0	0
$203$	3.43570	0.241139
$204$	0	0
$205$	5.09877	0.356114
$206$	0	0
$207$	27.1943	1.89014
$208$	0	0
$209$	−27.5180	−1.90346
$210$	0	0
$211$	−9.71649	−0.668911	−0.334455	−	0.942412i	$-0.608552\pi$
$211$	−9.71649	−0.668911	−0.334455	+	0.942412i	$0.608552\pi$
$212$	0	0
$213$	20.8255	1.42694
$214$	0	0
$215$	2.48390	0.169401
$216$	0	0
$217$	0.342534	0.0232527
$218$	0	0
$219$	−13.8385	−0.935116
$220$	0	0
$221$	4.03238	0.271247
$222$	0	0
$223$	4.27377	0.286193	0.143096	−	0.989709i	$-0.454294\pi$
$223$	4.27377	0.286193	0.143096	+	0.989709i	$0.454294\pi$
$224$	0	0
$225$	−23.4335	−1.56223
$226$	0	0
$227$	1.37427	0.0912133	0.0456067	−	0.998959i	$-0.485478\pi$
$227$	1.37427	0.0912133	0.0456067	+	0.998959i	$0.485478\pi$
$228$	0	0
$229$	−10.3414	−0.683378	−0.341689	−	0.939813i	$-0.610999\pi$
$229$	−10.3414	−0.683378	−0.341689	+	0.939813i	$0.610999\pi$
$230$	0	0
$231$	−25.9455	−1.70709
$232$	0	0
$233$	6.14073	0.402293	0.201146	−	0.979561i	$-0.435533\pi$
$233$	6.14073	0.402293	0.201146	+	0.979561i	$0.435533\pi$
$234$	0	0
$235$	1.84480	0.120341
$236$	0	0
$237$	22.3089	1.44912
$238$	0	0
$239$	5.79965	0.375148	0.187574	−	0.982250i	$-0.439937\pi$
$239$	5.79965	0.375148	0.187574	+	0.982250i	$0.439937\pi$
$240$	0	0
$241$	−21.4392	−1.38102	−0.690509	−	0.723324i	$-0.742613\pi$
$241$	−21.4392	−1.38102	−0.690509	+	0.723324i	$0.742613\pi$
$242$	0	0
$243$	15.2469	0.978089
$244$	0	0
$245$	1.35151	0.0863446
$246$	0	0
$247$	−3.54557	−0.225599
$248$	0	0
$249$	21.5471	1.36549
$250$	0	0
$251$	−14.2959	−0.902351	−0.451176	−	0.892435i	$-0.648995\pi$
$251$	−14.2959	−0.902351	−0.451176	+	0.892435i	$0.648995\pi$
$252$	0	0
$253$	−25.5711	−1.60764
$254$	0	0
$255$	−8.71600	−0.545817
$256$	0	0
$257$	−3.35974	−0.209575	−0.104788	−	0.994495i	$-0.533416\pi$
$257$	−3.35974	−0.209575	−0.104788	+	0.994495i	$0.533416\pi$
$258$	0	0
$259$	−5.97577	−0.371317
$260$	0	0
$261$	−8.36320	−0.517669
$262$	0	0
$263$	−16.7278	−1.03148	−0.515741	−	0.856744i	$-0.672483\pi$
$263$	−16.7278	−1.03148	−0.515741	+	0.856744i	$0.672483\pi$
$264$	0	0
$265$	3.57461	0.219587
$266$	0	0
$267$	−6.86741	−0.420278
$268$	0	0
$269$	2.95558	0.180205	0.0901026	−	0.995932i	$-0.471281\pi$
$269$	2.95558	0.180205	0.0901026	+	0.995932i	$0.471281\pi$
$270$	0	0
$271$	−0.610591	−0.0370907	−0.0185454	−	0.999828i	$-0.505904\pi$
$271$	−0.610591	−0.0370907	−0.0185454	+	0.999828i	$0.505904\pi$
$272$	0	0
$273$	−3.34295	−0.202324
$274$	0	0
$275$	22.0348	1.32875
$276$	0	0
$277$	17.0405	1.02387	0.511933	−	0.859025i	$-0.328930\pi$
$277$	17.0405	1.02387	0.511933	+	0.859025i	$0.328930\pi$
$278$	0	0
$279$	−0.833799	−0.0499183
$280$	0	0
$281$	−6.02204	−0.359245	−0.179622	−	0.983736i	$-0.557488\pi$
$281$	−6.02204	−0.359245	−0.179622	+	0.983736i	$0.557488\pi$
$282$	0	0
$283$	30.3910	1.80656	0.903279	−	0.429053i	$-0.141153\pi$
$283$	30.3910	1.80656	0.903279	+	0.429053i	$0.141153\pi$
$284$	0	0
$285$	7.66376	0.453962
$286$	0	0
$287$	22.4672	1.32620
$288$	0	0
$289$	29.3269	1.72511
$290$	0	0
$291$	−17.9595	−1.05281
$292$	0	0
$293$	27.2381	1.59127	0.795634	−	0.605778i	$-0.207138\pi$
$293$	27.2381	1.59127	0.795634	+	0.605778i	$0.207138\pi$
$294$	0	0
$295$	3.59589	0.209361
$296$	0	0
$297$	24.4099	1.41641
$298$	0	0
$299$	−3.29472	−0.190538
$300$	0	0
$301$	10.9451	0.630863
$302$	0	0
$303$	6.58505	0.378301
$304$	0	0
$305$	1.09429	0.0626587
$306$	0	0
$307$	−3.27980	−0.187188	−0.0935939	−	0.995610i	$-0.529836\pi$
$307$	−3.27980	−0.187188	−0.0935939	+	0.995610i	$0.529836\pi$
$308$	0	0
$309$	−14.9624	−0.851184
$310$	0	0
$311$	1.59478	0.0904319	0.0452160	−	0.998977i	$-0.485602\pi$
$311$	1.59478	0.0904319	0.0452160	+	0.998977i	$0.485602\pi$
$312$	0	0
$313$	−5.97937	−0.337974	−0.168987	−	0.985618i	$-0.554050\pi$
$313$	−5.97937	−0.337974	−0.168987	+	0.985618i	$0.554050\pi$
$314$	0	0
$315$	4.47833	0.252325
$316$	0	0
$317$	9.79594	0.550195	0.275098	−	0.961416i	$-0.411290\pi$
$317$	9.79594	0.550195	0.275098	+	0.961416i	$0.411290\pi$
$318$	0	0
$319$	7.86401	0.440300
$320$	0	0
$321$	10.8192	0.603871
$322$	0	0
$323$	−40.7340	−2.26650
$324$	0	0
$325$	2.83907	0.157483
$326$	0	0
$327$	28.2897	1.56442
$328$	0	0
$329$	8.12893	0.448162
$330$	0	0
$331$	19.3796	1.06520	0.532600	−	0.846367i	$-0.321215\pi$
$331$	19.3796	1.06520	0.532600	+	0.846367i	$0.321215\pi$
$332$	0	0
$333$	14.5463	0.797131
$334$	0	0
$335$	−1.51184	−0.0826005
$336$	0	0
$337$	5.60858	0.305519	0.152759	−	0.988263i	$-0.451184\pi$
$337$	5.60858	0.305519	0.152759	+	0.988263i	$0.451184\pi$
$338$	0	0
$339$	−2.96776	−0.161187
$340$	0	0
$341$	0.784031	0.0424576
$342$	0	0
$343$	20.0172	1.08083
$344$	0	0
$345$	7.12154	0.383411
$346$	0	0
$347$	32.2656	1.73211	0.866053	−	0.499952i	$-0.166649\pi$
$347$	32.2656	1.73211	0.866053	+	0.499952i	$0.166649\pi$
$348$	0	0
$349$	−1.85723	−0.0994154	−0.0497077	−	0.998764i	$-0.515829\pi$
$349$	−1.85723	−0.0994154	−0.0497077	+	0.998764i	$0.515829\pi$
$350$	0	0
$351$	3.14510	0.167873
$352$	0	0
$353$	−34.0693	−1.81332	−0.906662	−	0.421858i	$-0.861378\pi$
$353$	−34.0693	−1.81332	−0.906662	+	0.421858i	$0.861378\pi$
$354$	0	0
$355$	3.38004	0.179394
$356$	0	0
$357$	−38.4062	−2.03267
$358$	0	0
$359$	13.1396	0.693481	0.346741	−	0.937961i	$-0.387288\pi$
$359$	13.1396	0.693481	0.346741	+	0.937961i	$0.387288\pi$
$360$	0	0
$361$	16.8164	0.885072
$362$	0	0
$363$	−28.4890	−1.49528
$364$	0	0
$365$	−2.24602	−0.117562
$366$	0	0
$367$	13.8183	0.721310	0.360655	−	0.932699i	$-0.382553\pi$
$367$	13.8183	0.721310	0.360655	+	0.932699i	$0.382553\pi$
$368$	0	0
$369$	−54.6899	−2.84704
$370$	0	0
$371$	15.7512	0.817760
$372$	0	0
$373$	−18.2939	−0.947222	−0.473611	−	0.880734i	$-0.657050\pi$
$373$	−18.2939	−0.947222	−0.473611	+	0.880734i	$0.657050\pi$
$374$	0	0
$375$	−12.5395	−0.647536
$376$	0	0
$377$	1.01324	0.0521845
$378$	0	0
$379$	−7.47786	−0.384112	−0.192056	−	0.981384i	$-0.561516\pi$
$379$	−7.47786	−0.384112	−0.192056	+	0.981384i	$0.561516\pi$
$380$	0	0
$381$	−32.3367	−1.65666
$382$	0	0
$383$	36.7930	1.88004	0.940018	−	0.341124i	$-0.110808\pi$
$383$	36.7930	1.88004	0.940018	+	0.341124i	$0.110808\pi$
$384$	0	0
$385$	−4.21103	−0.214614
$386$	0	0
$387$	−26.6426	−1.35432
$388$	0	0
$389$	−4.23814	−0.214882	−0.107441	−	0.994211i	$-0.534266\pi$
$389$	−4.23814	−0.214882	−0.107441	+	0.994211i	$0.534266\pi$
$390$	0	0
$391$	−37.8521	−1.91426
$392$	0	0
$393$	14.6009	0.736519
$394$	0	0
$395$	3.62080	0.182182
$396$	0	0
$397$	23.0976	1.15923	0.579617	−	0.814889i	$-0.303202\pi$
$397$	23.0976	1.15923	0.579617	+	0.814889i	$0.303202\pi$
$398$	0	0
$399$	33.7696	1.69059
$400$	0	0
$401$	−13.2705	−0.662697	−0.331349	−	0.943508i	$-0.607504\pi$
$401$	−13.2705	−0.662697	−0.331349	+	0.943508i	$0.607504\pi$
$402$	0	0
$403$	0.101019	0.00503209
$404$	0	0
$405$	−0.110245	−0.00547813
$406$	0	0
$407$	−13.6780	−0.677995
$408$	0	0
$409$	17.6653	0.873495	0.436748	−	0.899584i	$-0.356130\pi$
$409$	17.6653	0.873495	0.436748	+	0.899584i	$0.356130\pi$
$410$	0	0
$411$	18.4783	0.911470
$412$	0	0
$413$	15.8449	0.779679
$414$	0	0
$415$	3.49715	0.171668
$416$	0	0
$417$	−15.6450	−0.766139
$418$	0	0
$419$	−17.9026	−0.874600	−0.437300	−	0.899316i	$-0.644065\pi$
$419$	−17.9026	−0.874600	−0.437300	+	0.899316i	$0.644065\pi$
$420$	0	0
$421$	8.13471	0.396461	0.198231	−	0.980155i	$-0.436480\pi$
$421$	8.13471	0.396461	0.198231	+	0.980155i	$0.436480\pi$
$422$	0	0
$423$	−19.7875	−0.962101
$424$	0	0
$425$	32.6173	1.58217
$426$	0	0
$427$	4.82186	0.233346
$428$	0	0
$429$	−7.65172	−0.369428
$430$	0	0
$431$	26.2484	1.26434	0.632171	−	0.774829i	$-0.282164\pi$
$431$	26.2484	1.26434	0.632171	+	0.774829i	$0.282164\pi$
$432$	0	0
$433$	−40.7822	−1.95987	−0.979933	−	0.199328i	$-0.936124\pi$
$433$	−40.7822	−1.95987	−0.979933	+	0.199328i	$0.936124\pi$
$434$	0	0
$435$	−2.19012	−0.105008
$436$	0	0
$437$	33.2823	1.59211
$438$	0	0
$439$	35.7848	1.70792	0.853958	−	0.520343i	$-0.174196\pi$
$439$	35.7848	1.70792	0.853958	+	0.520343i	$0.174196\pi$
$440$	0	0
$441$	−14.4964	−0.690304
$442$	0	0
$443$	−7.36254	−0.349805	−0.174902	−	0.984586i	$-0.555961\pi$
$443$	−7.36254	−0.349805	−0.174902	+	0.984586i	$0.555961\pi$
$444$	0	0
$445$	−1.11460	−0.0528371
$446$	0	0
$447$	32.9742	1.55963
$448$	0	0
$449$	−4.99015	−0.235500	−0.117750	−	0.993043i	$-0.537568\pi$
$449$	−4.99015	−0.235500	−0.117750	+	0.993043i	$0.537568\pi$
$450$	0	0
$451$	51.4255	2.42153
$452$	0	0
$453$	3.92675	0.184495
$454$	0	0
$455$	−0.542570	−0.0254361
$456$	0	0
$457$	22.9918	1.07551	0.537756	−	0.843100i	$-0.319272\pi$
$457$	22.9918	1.07551	0.537756	+	0.843100i	$0.319272\pi$
$458$	0	0
$459$	36.1332	1.68655
$460$	0	0
$461$	−0.0673800	−0.00313820	−0.00156910	−	0.999999i	$-0.500499\pi$
$461$	−0.0673800	−0.00313820	−0.00156910	+	0.999999i	$0.500499\pi$
$462$	0	0
$463$	19.5937	0.910597	0.455299	−	0.890339i	$-0.349533\pi$
$463$	19.5937	0.910597	0.455299	+	0.890339i	$0.349533\pi$
$464$	0	0
$465$	−0.218352	−0.0101258
$466$	0	0
$467$	−12.0880	−0.559366	−0.279683	−	0.960092i	$-0.590229\pi$
$467$	−12.0880	−0.559366	−0.279683	+	0.960092i	$0.590229\pi$
$468$	0	0
$469$	−6.66176	−0.307611
$470$	0	0
$471$	43.3956	1.99956
$472$	0	0
$473$	25.0523	1.15191
$474$	0	0
$475$	−28.6795	−1.31591
$476$	0	0
$477$	−38.3416	−1.75554
$478$	0	0
$479$	10.1682	0.464596	0.232298	−	0.972645i	$-0.425375\pi$
$479$	10.1682	0.464596	0.232298	+	0.972645i	$0.425375\pi$
$480$	0	0
$481$	−1.76235	−0.0803561
$482$	0	0
$483$	31.3804	1.42786
$484$	0	0
$485$	−2.91488	−0.132358
$486$	0	0
$487$	31.0619	1.40755	0.703774	−	0.710424i	$-0.251497\pi$
$487$	31.0619	1.40755	0.703774	+	0.710424i	$0.251497\pi$
$488$	0	0
$489$	27.1568	1.22807
$490$	0	0
$491$	−25.1521	−1.13510	−0.567548	−	0.823340i	$-0.692108\pi$
$491$	−25.1521	−1.13510	−0.567548	+	0.823340i	$0.692108\pi$
$492$	0	0
$493$	11.6408	0.524276
$494$	0	0
$495$	10.2505	0.460726
$496$	0	0
$497$	14.8938	0.668078
$498$	0	0
$499$	−11.7282	−0.525028	−0.262514	−	0.964928i	$-0.584552\pi$
$499$	−11.7282	−0.525028	−0.262514	+	0.964928i	$0.584552\pi$
$500$	0	0
$501$	−12.9994	−0.580772
$502$	0	0
$503$	34.0027	1.51611	0.758053	−	0.652193i	$-0.226151\pi$
$503$	34.0027	1.51611	0.758053	+	0.652193i	$0.226151\pi$
$504$	0	0
$505$	1.06877	0.0475598
$506$	0	0
$507$	35.5299	1.57794
$508$	0	0
$509$	8.81652	0.390785	0.195393	−	0.980725i	$-0.437402\pi$
$509$	8.81652	0.390785	0.195393	+	0.980725i	$0.437402\pi$
$510$	0	0
$511$	−9.89687	−0.437812
$512$	0	0
$513$	−31.7710	−1.40272
$514$	0	0
$515$	−2.42845	−0.107010
$516$	0	0
$517$	18.6064	0.818309
$518$	0	0
$519$	55.4215	2.43274
$520$	0	0
$521$	−4.41238	−0.193310	−0.0966549	−	0.995318i	$-0.530814\pi$
$521$	−4.41238	−0.193310	−0.0966549	+	0.995318i	$0.530814\pi$
$522$	0	0
$523$	−3.71051	−0.162249	−0.0811247	−	0.996704i	$-0.525851\pi$
$523$	−3.71051	−0.162249	−0.0811247	+	0.996704i	$0.525851\pi$
$524$	0	0
$525$	−27.0406	−1.18015
$526$	0	0
$527$	1.16057	0.0505554
$528$	0	0
$529$	7.92760	0.344678
$530$	0	0
$531$	−38.5699	−1.67379
$532$	0	0
$533$	6.62593	0.287001
$534$	0	0
$535$	1.75599	0.0759182
$536$	0	0
$537$	32.9821	1.42328
$538$	0	0
$539$	13.6311	0.587134
$540$	0	0
$541$	4.50768	0.193800	0.0969002	−	0.995294i	$-0.469107\pi$
$541$	4.50768	0.193800	0.0969002	+	0.995294i	$0.469107\pi$
$542$	0	0
$543$	−23.8499	−1.02350
$544$	0	0
$545$	4.59149	0.196678
$546$	0	0
$547$	11.8748	0.507730	0.253865	−	0.967240i	$-0.418298\pi$
$547$	11.8748	0.507730	0.253865	+	0.967240i	$0.418298\pi$
$548$	0	0
$549$	−11.7374	−0.500941
$550$	0	0
$551$	−10.2355	−0.436046
$552$	0	0
$553$	15.9547	0.678463
$554$	0	0
$555$	3.80932	0.161697
$556$	0	0
$557$	−29.0077	−1.22909	−0.614547	−	0.788880i	$-0.710661\pi$
$557$	−29.0077	−1.22909	−0.614547	+	0.788880i	$0.710661\pi$
$558$	0	0
$559$	3.22787	0.136524
$560$	0	0
$561$	−87.9084	−3.71150
$562$	0	0
$563$	6.02949	0.254113	0.127056	−	0.991896i	$-0.459447\pi$
$563$	6.02949	0.254113	0.127056	+	0.991896i	$0.459447\pi$
$564$	0	0
$565$	−0.481676	−0.0202643
$566$	0	0
$567$	−0.485784	−0.0204010
$568$	0	0
$569$	15.0333	0.630227	0.315114	−	0.949054i	$-0.397957\pi$
$569$	15.0333	0.630227	0.315114	+	0.949054i	$0.397957\pi$
$570$	0	0
$571$	−30.7054	−1.28498	−0.642491	−	0.766294i	$-0.722099\pi$
$571$	−30.7054	−1.28498	−0.642491	+	0.766294i	$0.722099\pi$
$572$	0	0
$573$	−64.5997	−2.69869
$574$	0	0
$575$	−26.6504	−1.11140
$576$	0	0
$577$	21.9801	0.915044	0.457522	−	0.889198i	$-0.348737\pi$
$577$	21.9801	0.915044	0.457522	+	0.889198i	$0.348737\pi$
$578$	0	0
$579$	42.2670	1.75655
$580$	0	0
$581$	15.4099	0.639308
$582$	0	0
$583$	36.0531	1.49317
$584$	0	0
$585$	1.32073	0.0546054
$586$	0	0
$587$	−19.1862	−0.791898	−0.395949	−	0.918272i	$-0.629584\pi$
$587$	−19.1862	−0.791898	−0.395949	+	0.918272i	$0.629584\pi$
$588$	0	0
$589$	−1.02046	−0.0420474
$590$	0	0
$591$	23.8897	0.982692
$592$	0	0
$593$	37.4280	1.53698	0.768492	−	0.639859i	$-0.221007\pi$
$593$	37.4280	1.53698	0.768492	+	0.639859i	$0.221007\pi$
$594$	0	0
$595$	−6.23344	−0.255546
$596$	0	0
$597$	22.0076	0.900711
$598$	0	0
$599$	13.0922	0.534933	0.267467	−	0.963567i	$-0.413814\pi$
$599$	13.0922	0.534933	0.267467	+	0.963567i	$0.413814\pi$
$600$	0	0
$601$	13.4410	0.548269	0.274135	−	0.961691i	$-0.411609\pi$
$601$	13.4410	0.548269	0.274135	+	0.961691i	$0.411609\pi$
$602$	0	0
$603$	16.2161	0.660371
$604$	0	0
$605$	−4.62384	−0.187986
$606$	0	0
$607$	37.0351	1.50321	0.751604	−	0.659615i	$-0.229281\pi$
$607$	37.0351	1.50321	0.751604	+	0.659615i	$0.229281\pi$
$608$	0	0
$609$	−9.65055	−0.391060
$610$	0	0
$611$	2.39735	0.0969862
$612$	0	0
$613$	−10.9451	−0.442067	−0.221034	−	0.975266i	$-0.570943\pi$
$613$	−10.9451	−0.442067	−0.221034	+	0.975266i	$0.570943\pi$
$614$	0	0
$615$	−14.3220	−0.577518
$616$	0	0
$617$	9.15843	0.368705	0.184352	−	0.982860i	$-0.440981\pi$
$617$	9.15843	0.368705	0.184352	+	0.982860i	$0.440981\pi$
$618$	0	0
$619$	8.14167	0.327241	0.163621	−	0.986523i	$-0.447683\pi$
$619$	8.14167	0.327241	0.163621	+	0.986523i	$0.447683\pi$
$620$	0	0
$621$	−29.5232	−1.18472
$622$	0	0
$623$	−4.91137	−0.196770
$624$	0	0
$625$	21.9256	0.877024
$626$	0	0
$627$	77.2956	3.08689
$628$	0	0
$629$	−20.2471	−0.807305
$630$	0	0
$631$	−8.94570	−0.356123	−0.178061	−	0.984019i	$-0.556983\pi$
$631$	−8.94570	−0.356123	−0.178061	+	0.984019i	$0.556983\pi$
$632$	0	0
$633$	27.2927	1.08479
$634$	0	0
$635$	−5.24833	−0.208274
$636$	0	0
$637$	1.75630	0.0695873
$638$	0	0
$639$	−36.2546	−1.43421
$640$	0	0
$641$	41.2223	1.62818	0.814092	−	0.580735i	$-0.197235\pi$
$641$	41.2223	1.62818	0.814092	+	0.580735i	$0.197235\pi$
$642$	0	0
$643$	36.6787	1.44647	0.723233	−	0.690604i	$-0.242655\pi$
$643$	36.6787	1.44647	0.723233	+	0.690604i	$0.242655\pi$
$644$	0	0
$645$	−6.97705	−0.274721
$646$	0	0
$647$	8.95392	0.352015	0.176007	−	0.984389i	$-0.443682\pi$
$647$	8.95392	0.352015	0.176007	+	0.984389i	$0.443682\pi$
$648$	0	0
$649$	36.2677	1.42363
$650$	0	0
$651$	−0.962146	−0.0377095
$652$	0	0
$653$	36.4343	1.42579	0.712893	−	0.701273i	$-0.247385\pi$
$653$	36.4343	1.42579	0.712893	+	0.701273i	$0.247385\pi$
$654$	0	0
$655$	2.36977	0.0925947
$656$	0	0
$657$	24.0910	0.939881
$658$	0	0
$659$	−42.9491	−1.67306	−0.836529	−	0.547923i	$-0.815419\pi$
$659$	−42.9491	−1.67306	−0.836529	+	0.547923i	$0.815419\pi$
$660$	0	0
$661$	−25.1644	−0.978783	−0.489392	−	0.872064i	$-0.662781\pi$
$661$	−25.1644	−0.978783	−0.489392	+	0.872064i	$0.662781\pi$
$662$	0	0
$663$	−11.3266	−0.439888
$664$	0	0
$665$	5.48090	0.212540
$666$	0	0
$667$	−9.51131	−0.368279
$668$	0	0
$669$	−12.0046	−0.464126
$670$	0	0
$671$	11.0368	0.426072
$672$	0	0
$673$	−0.384731	−0.0148303	−0.00741514	−	0.999973i	$-0.502360\pi$
$673$	−0.384731	−0.0148303	−0.00741514	+	0.999973i	$0.502360\pi$
$674$	0	0
$675$	25.4402	0.979195
$676$	0	0
$677$	−7.24898	−0.278601	−0.139301	−	0.990250i	$-0.544485\pi$
$677$	−7.24898	−0.278601	−0.139301	+	0.990250i	$0.544485\pi$
$678$	0	0
$679$	−12.8442	−0.492913
$680$	0	0
$681$	−3.86019	−0.147923
$682$	0	0
$683$	−41.6758	−1.59468	−0.797340	−	0.603530i	$-0.793760\pi$
$683$	−41.6758	−1.59468	−0.797340	+	0.603530i	$0.793760\pi$
$684$	0	0
$685$	2.99909	0.114589
$686$	0	0
$687$	29.0480	1.10825
$688$	0	0
$689$	4.64526	0.176970
$690$	0	0
$691$	−19.8459	−0.754972	−0.377486	−	0.926015i	$-0.623211\pi$
$691$	−19.8459	−0.754972	−0.377486	+	0.926015i	$0.623211\pi$
$692$	0	0
$693$	45.1679	1.71579
$694$	0	0
$695$	−2.53923	−0.0963184
$696$	0	0
$697$	76.1234	2.88338
$698$	0	0
$699$	−17.2487	−0.652407
$700$	0	0
$701$	−24.5409	−0.926897	−0.463449	−	0.886124i	$-0.653388\pi$
$701$	−24.5409	−0.926897	−0.463449	+	0.886124i	$0.653388\pi$
$702$	0	0
$703$	17.8028	0.671444
$704$	0	0
$705$	−5.18187	−0.195161
$706$	0	0
$707$	4.70944	0.177117
$708$	0	0
$709$	3.38022	0.126947	0.0634734	−	0.997984i	$-0.479782\pi$
$709$	3.38022	0.126947	0.0634734	+	0.997984i	$0.479782\pi$
$710$	0	0
$711$	−38.8371	−1.45650
$712$	0	0
$713$	−0.948264	−0.0355128
$714$	0	0
$715$	−1.24190	−0.0464443
$716$	0	0
$717$	−16.2907	−0.608387
$718$	0	0
$719$	−13.2636	−0.494650	−0.247325	−	0.968933i	$-0.579552\pi$
$719$	−13.2636	−0.494650	−0.247325	+	0.968933i	$0.579552\pi$
$720$	0	0
$721$	−10.7007	−0.398515
$722$	0	0
$723$	60.2206	2.23963
$724$	0	0
$725$	8.19594	0.304389
$726$	0	0
$727$	−19.5668	−0.725694	−0.362847	−	0.931849i	$-0.618195\pi$
$727$	−19.5668	−0.725694	−0.362847	+	0.931849i	$0.618195\pi$
$728$	0	0
$729$	−43.5526	−1.61306
$730$	0	0
$731$	37.0841	1.37160
$732$	0	0
$733$	−32.5815	−1.20342	−0.601712	−	0.798713i	$-0.705515\pi$
$733$	−32.5815	−1.20342	−0.601712	+	0.798713i	$0.705515\pi$
$734$	0	0
$735$	−3.79626	−0.140027
$736$	0	0
$737$	−15.2482	−0.561674
$738$	0	0
$739$	−21.2341	−0.781110	−0.390555	−	0.920580i	$-0.627717\pi$
$739$	−21.2341	−0.781110	−0.390555	+	0.920580i	$0.627717\pi$
$740$	0	0
$741$	9.95917	0.365859
$742$	0	0
$743$	−6.49461	−0.238264	−0.119132	−	0.992878i	$-0.538011\pi$
$743$	−6.49461	−0.238264	−0.119132	+	0.992878i	$0.538011\pi$
$744$	0	0
$745$	5.35181	0.196075
$746$	0	0
$747$	−37.5108	−1.37245
$748$	0	0
$749$	7.73761	0.282726
$750$	0	0
$751$	−24.3185	−0.887396	−0.443698	−	0.896176i	$-0.646334\pi$
$751$	−24.3185	−0.887396	−0.443698	+	0.896176i	$0.646334\pi$
$752$	0	0
$753$	40.1559	1.46336
$754$	0	0
$755$	0.637322	0.0231945
$756$	0	0
$757$	29.5600	1.07438	0.537188	−	0.843462i	$-0.319486\pi$
$757$	29.5600	1.07438	0.537188	+	0.843462i	$0.319486\pi$
$758$	0	0
$759$	71.8269	2.60715
$760$	0	0
$761$	1.36655	0.0495373	0.0247687	−	0.999693i	$-0.492115\pi$
$761$	1.36655	0.0495373	0.0247687	+	0.999693i	$0.492115\pi$
$762$	0	0
$763$	20.2320	0.732446
$764$	0	0
$765$	15.1735	0.548598
$766$	0	0
$767$	4.67292	0.168729
$768$	0	0
$769$	−7.47293	−0.269481	−0.134740	−	0.990881i	$-0.543020\pi$
$769$	−7.47293	−0.269481	−0.134740	+	0.990881i	$0.543020\pi$
$770$	0	0
$771$	9.43721	0.339873
$772$	0	0
$773$	−7.98918	−0.287351	−0.143676	−	0.989625i	$-0.545892\pi$
$773$	−7.98918	−0.287351	−0.143676	+	0.989625i	$0.545892\pi$
$774$	0	0
$775$	0.817123	0.0293519
$776$	0	0
$777$	16.7854	0.602173
$778$	0	0
$779$	−66.9334	−2.39814
$780$	0	0
$781$	34.0906	1.21986
$782$	0	0
$783$	9.07940	0.324471
$784$	0	0
$785$	7.04323	0.251383
$786$	0	0
$787$	−26.0664	−0.929165	−0.464583	−	0.885530i	$-0.653796\pi$
$787$	−26.0664	−0.929165	−0.464583	+	0.885530i	$0.653796\pi$
$788$	0	0
$789$	46.9869	1.67278
$790$	0	0
$791$	−2.12246	−0.0754659
$792$	0	0
$793$	1.42204	0.0504982
$794$	0	0
$795$	−10.0408	−0.356109
$796$	0	0
$797$	18.0531	0.639472	0.319736	−	0.947507i	$-0.396406\pi$
$797$	18.0531	0.639472	0.319736	+	0.947507i	$0.396406\pi$
$798$	0	0
$799$	27.5424	0.974381
$800$	0	0
$801$	11.9553	0.422420
$802$	0	0
$803$	−22.6531	−0.799410
$804$	0	0
$805$	5.09312	0.179509
$806$	0	0
$807$	−8.30196	−0.292243
$808$	0	0
$809$	−30.0267	−1.05568	−0.527842	−	0.849343i	$-0.676999\pi$
$809$	−30.0267	−1.05568	−0.527842	+	0.849343i	$0.676999\pi$
$810$	0	0
$811$	−19.3145	−0.678225	−0.339113	−	0.940746i	$-0.610127\pi$
$811$	−19.3145	−0.678225	−0.339113	+	0.940746i	$0.610127\pi$
$812$	0	0
$813$	1.71509	0.0601509
$814$	0	0
$815$	4.40763	0.154393
$816$	0	0
$817$	−32.6071	−1.14078
$818$	0	0
$819$	5.81966	0.203355
$820$	0	0
$821$	−17.6210	−0.614977	−0.307489	−	0.951552i	$-0.599489\pi$
$821$	−17.6210	−0.614977	−0.307489	+	0.951552i	$0.599489\pi$
$822$	0	0
$823$	−18.9876	−0.661865	−0.330933	−	0.943654i	$-0.607363\pi$
$823$	−18.9876	−0.661865	−0.330933	+	0.943654i	$0.607363\pi$
$824$	0	0
$825$	−61.8936	−2.15486
$826$	0	0
$827$	−8.88342	−0.308907	−0.154453	−	0.988000i	$-0.549362\pi$
$827$	−8.88342	−0.308907	−0.154453	+	0.988000i	$0.549362\pi$
$828$	0	0
$829$	−35.3730	−1.22855	−0.614277	−	0.789091i	$-0.710552\pi$
$829$	−35.3730	−1.22855	−0.614277	+	0.789091i	$0.710552\pi$
$830$	0	0
$831$	−47.8653	−1.66043
$832$	0	0
$833$	20.1777	0.699115
$834$	0	0
$835$	−2.10984	−0.0730142
$836$	0	0
$837$	0.905203	0.0312884
$838$	0	0
$839$	25.4271	0.877842	0.438921	−	0.898526i	$-0.355361\pi$
$839$	25.4271	0.877842	0.438921	+	0.898526i	$0.355361\pi$
$840$	0	0
$841$	−26.0749	−0.899136
$842$	0	0
$843$	16.9154	0.582596
$844$	0	0
$845$	5.76661	0.198377
$846$	0	0
$847$	−20.3745	−0.700076
$848$	0	0
$849$	−85.3656	−2.92974
$850$	0	0
$851$	16.5432	0.567094
$852$	0	0
$853$	−38.0283	−1.30206	−0.651032	−	0.759050i	$-0.725664\pi$
$853$	−38.0283	−1.30206	−0.651032	+	0.759050i	$0.725664\pi$
$854$	0	0
$855$	−13.3417	−0.456275
$856$	0	0
$857$	47.7529	1.63121	0.815604	−	0.578610i	$-0.196405\pi$
$857$	47.7529	1.63121	0.815604	+	0.578610i	$0.196405\pi$
$858$	0	0
$859$	6.83935	0.233356	0.116678	−	0.993170i	$-0.462776\pi$
$859$	6.83935	0.233356	0.116678	+	0.993170i	$0.462776\pi$
$860$	0	0
$861$	−63.1084	−2.15073
$862$	0	0
$863$	−26.4429	−0.900125	−0.450063	−	0.892997i	$-0.648598\pi$
$863$	−26.4429	−0.900125	−0.450063	+	0.892997i	$0.648598\pi$
$864$	0	0
$865$	8.99508	0.305842
$866$	0	0
$867$	−82.3764	−2.79765
$868$	0	0
$869$	36.5189	1.23882
$870$	0	0
$871$	−1.96466	−0.0665698
$872$	0	0
$873$	31.2653	1.05817
$874$	0	0
$875$	−8.96788	−0.303170
$876$	0	0
$877$	46.1572	1.55862	0.779309	−	0.626640i	$-0.215570\pi$
$877$	46.1572	1.55862	0.779309	+	0.626640i	$0.215570\pi$
$878$	0	0
$879$	−76.5094	−2.58060
$880$	0	0
$881$	50.5841	1.70422	0.852110	−	0.523362i	$-0.175323\pi$
$881$	50.5841	1.70422	0.852110	+	0.523362i	$0.175323\pi$
$882$	0	0
$883$	−35.7405	−1.20276	−0.601382	−	0.798961i	$-0.705383\pi$
$883$	−35.7405	−1.20276	−0.601382	+	0.798961i	$0.705383\pi$
$884$	0	0
$885$	−10.1005	−0.339526
$886$	0	0
$887$	34.3192	1.15233	0.576163	−	0.817335i	$-0.304549\pi$
$887$	34.3192	1.15233	0.576163	+	0.817335i	$0.304549\pi$
$888$	0	0
$889$	−23.1262	−0.775629
$890$	0	0
$891$	−1.11192	−0.0372507
$892$	0	0
$893$	−24.2173	−0.810402
$894$	0	0
$895$	5.35310	0.178934
$896$	0	0
$897$	9.25455	0.309000
$898$	0	0
$899$	0.291624	0.00972621
$900$	0	0
$901$	53.3681	1.77795
$902$	0	0
$903$	−30.7437	−1.02309
$904$	0	0
$905$	−3.87090	−0.128673
$906$	0	0
$907$	−8.69980	−0.288872	−0.144436	−	0.989514i	$-0.546137\pi$
$907$	−8.69980	−0.288872	−0.144436	+	0.989514i	$0.546137\pi$
$908$	0	0
$909$	−11.4638	−0.380229
$910$	0	0
$911$	−25.4294	−0.842513	−0.421257	−	0.906942i	$-0.638411\pi$
$911$	−25.4294	−0.842513	−0.421257	+	0.906942i	$0.638411\pi$
$912$	0	0
$913$	35.2718	1.16733
$914$	0	0
$915$	−3.07375	−0.101615
$916$	0	0
$917$	10.4422	0.344831
$918$	0	0
$919$	11.2219	0.370176	0.185088	−	0.982722i	$-0.440743\pi$
$919$	11.2219	0.370176	0.185088	+	0.982722i	$0.440743\pi$
$920$	0	0
$921$	9.21264	0.303567
$922$	0	0
$923$	4.39241	0.144578
$924$	0	0
$925$	−14.2554	−0.468713
$926$	0	0
$927$	26.0478	0.855521
$928$	0	0
$929$	−12.1441	−0.398435	−0.199217	−	0.979955i	$-0.563840\pi$
$929$	−12.1441	−0.398435	−0.199217	+	0.979955i	$0.563840\pi$
$930$	0	0
$931$	−17.7417	−0.581461
$932$	0	0
$933$	−4.47960	−0.146656
$934$	0	0
$935$	−14.2678	−0.466607
$936$	0	0
$937$	18.6238	0.608411	0.304206	−	0.952606i	$-0.401609\pi$
$937$	18.6238	0.608411	0.304206	+	0.952606i	$0.401609\pi$
$938$	0	0
$939$	16.7955	0.548101
$940$	0	0
$941$	19.3575	0.631035	0.315517	−	0.948920i	$-0.397822\pi$
$941$	19.3575	0.631035	0.315517	+	0.948920i	$0.397822\pi$
$942$	0	0
$943$	−62.1978	−2.02544
$944$	0	0
$945$	−4.86184	−0.158156
$946$	0	0
$947$	−15.4808	−0.503058	−0.251529	−	0.967850i	$-0.580933\pi$
$947$	−15.4808	−0.503058	−0.251529	+	0.967850i	$0.580933\pi$
$948$	0	0
$949$	−2.91874	−0.0947463
$950$	0	0
$951$	−27.5159	−0.892264
$952$	0	0
$953$	−56.4856	−1.82975	−0.914874	−	0.403739i	$-0.867710\pi$
$953$	−56.4856	−1.82975	−0.914874	+	0.403739i	$0.867710\pi$
$954$	0	0
$955$	−10.4847	−0.339278
$956$	0	0
$957$	−22.0893	−0.714045
$958$	0	0
$959$	13.2152	0.426741
$960$	0	0
$961$	−30.9709	−0.999062
$962$	0	0
$963$	−18.8350	−0.606948
$964$	0	0
$965$	6.86005	0.220833
$966$	0	0
$967$	−44.2085	−1.42165	−0.710825	−	0.703369i	$-0.751678\pi$
$967$	−44.2085	−1.42165	−0.710825	+	0.703369i	$0.751678\pi$
$968$	0	0
$969$	114.418	3.67564
$970$	0	0
$971$	−15.9609	−0.512211	−0.256105	−	0.966649i	$-0.582439\pi$
$971$	−15.9609	−0.512211	−0.256105	+	0.966649i	$0.582439\pi$
$972$	0	0
$973$	−11.1889	−0.358698
$974$	0	0
$975$	−7.97469	−0.255394
$976$	0	0
$977$	−0.168109	−0.00537829	−0.00268914	−	0.999996i	$-0.500856\pi$
$977$	−0.168109	−0.00537829	−0.00268914	+	0.999996i	$0.500856\pi$
$978$	0	0
$979$	−11.2417	−0.359286
$980$	0	0
$981$	−49.2488	−1.57239
$982$	0	0
$983$	−57.8922	−1.84647	−0.923236	−	0.384232i	$-0.874466\pi$
$983$	−57.8922	−1.84647	−0.923236	+	0.384232i	$0.874466\pi$
$984$	0	0
$985$	3.87737	0.123543
$986$	0	0
$987$	−22.8334	−0.726795
$988$	0	0
$989$	−30.3001	−0.963487
$990$	0	0
$991$	−29.0488	−0.922766	−0.461383	−	0.887201i	$-0.652647\pi$
$991$	−29.0488	−0.922766	−0.461383	+	0.887201i	$0.652647\pi$
$992$	0	0
$993$	−54.4355	−1.72746
$994$	0	0
$995$	3.57190	0.113237
$996$	0	0
$997$	−26.6819	−0.845024	−0.422512	−	0.906357i	$-0.638852\pi$
$997$	−26.6819	−0.845024	−0.422512	+	0.906357i	$0.638852\pi$
$998$	0	0
$999$	−15.7920	−0.499636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8012.2.a.b.1.6	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8012.2.a.b.1.6	✓	88	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.80891 q^{3} -0.455894 q^{5} -2.00885 q^{7} +4.88996 q^{9} +O(q^{10})\)	\(q-2.80891 q^{3} -0.455894 q^{5} -2.00885 q^{7} +4.88996 q^{9} -4.59808 q^{11} -0.592441 q^{13} +1.28056 q^{15} -6.80638 q^{17} +5.98468 q^{19} +5.64267 q^{21} +5.56126 q^{23} -4.79216 q^{25} -5.30872 q^{27} -1.71028 q^{29} -0.170513 q^{31} +12.9156 q^{33} +0.915822 q^{35} +2.97472 q^{37} +1.66411 q^{39} -11.1841 q^{41} -5.44843 q^{43} -2.22930 q^{45} -4.04656 q^{47} -2.96452 q^{49} +19.1185 q^{51} -7.84089 q^{53} +2.09624 q^{55} -16.8104 q^{57} -7.88757 q^{59} -2.40031 q^{61} -9.82319 q^{63} +0.270090 q^{65} +3.31621 q^{67} -15.6211 q^{69} -7.41409 q^{71} +4.92664 q^{73} +13.4607 q^{75} +9.23686 q^{77} -7.94220 q^{79} +0.241822 q^{81} -7.67098 q^{83} +3.10299 q^{85} +4.80402 q^{87} +2.44487 q^{89} +1.19012 q^{91} +0.478954 q^{93} -2.72838 q^{95} +6.39378 q^{97} -22.4844 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) \) 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9} + 16 q^{11} - 3 q^{13} + 21 q^{15} + 32 q^{17} + 49 q^{19} + 7 q^{21} + 24 q^{23} + 114 q^{25} + 82 q^{27} + 2 q^{29} + 40 q^{31} + 31 q^{33} + 26 q^{35} + 37 q^{39} + 22 q^{41} + 98 q^{43} + 12 q^{45} + 48 q^{47} + 132 q^{49} + 55 q^{51} - q^{53} + 116 q^{55} + 62 q^{57} + 34 q^{59} + 132 q^{63} + 39 q^{65} + 75 q^{67} + 15 q^{69} + 24 q^{71} + 104 q^{73} + 87 q^{75} + 4 q^{77} + 111 q^{79} + 128 q^{81} + 64 q^{83} + 7 q^{85} + 115 q^{87} + 14 q^{89} + 73 q^{91} - 7 q^{93} + 51 q^{95} + 117 q^{97} + 72 q^{99}+O(q^{100}) \) 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9 + 16 * q^11 - 3 * q^13 + 21 * q^15 + 32 * q^17 + 49 * q^19 + 7 * q^21 + 24 * q^23 + 114 * q^25 + 82 * q^27 + 2 * q^29 + 40 * q^31 + 31 * q^33 + 26 * q^35 + 37 * q^39 + 22 * q^41 + 98 * q^43 + 12 * q^45 + 48 * q^47 + 132 * q^49 + 55 * q^51 - q^53 + 116 * q^55 + 62 * q^57 + 34 * q^59 + 132 * q^63 + 39 * q^65 + 75 * q^67 + 15 * q^69 + 24 * q^71 + 104 * q^73 + 87 * q^75 + 4 * q^77 + 111 * q^79 + 128 * q^81 + 64 * q^83 + 7 * q^85 + 115 * q^87 + 14 * q^89 + 73 * q^91 - 7 * q^93 + 51 * q^95 + 117 * q^97 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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