LMFDB - Embedded newform 8012.2.a.b.1.17

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			1.17
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)

$f(q)$	$=$	$q-2.04818 q^{3} +1.65769 q^{5} -0.109887 q^{7} +1.19506 q^{9} +O(q^{10})$	$q-2.04818 q^{3} +1.65769 q^{5} -0.109887 q^{7} +1.19506 q^{9} +1.18302 q^{11} +0.0953931 q^{13} -3.39526 q^{15} +4.88654 q^{17} +8.36455 q^{19} +0.225068 q^{21} -8.76002 q^{23} -2.25205 q^{25} +3.69686 q^{27} +6.50912 q^{29} +3.01815 q^{31} -2.42303 q^{33} -0.182159 q^{35} +1.27668 q^{37} -0.195383 q^{39} -10.4931 q^{41} +9.50239 q^{43} +1.98104 q^{45} +10.6601 q^{47} -6.98792 q^{49} -10.0085 q^{51} -3.94211 q^{53} +1.96108 q^{55} -17.1321 q^{57} -4.33550 q^{59} +7.35520 q^{61} -0.131321 q^{63} +0.158133 q^{65} +9.54182 q^{67} +17.9421 q^{69} +2.52756 q^{71} +11.4899 q^{73} +4.61261 q^{75} -0.129998 q^{77} +0.886677 q^{79} -11.1570 q^{81} -14.4973 q^{83} +8.10039 q^{85} -13.3319 q^{87} -7.41900 q^{89} -0.0104824 q^{91} -6.18172 q^{93} +13.8659 q^{95} -11.4811 q^{97} +1.41377 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.04818	−1.18252	−0.591260	−	0.806481i	$-0.701369\pi$
$3$	−2.04818	−1.18252	−0.591260	+	0.806481i	$0.701369\pi$
$4$	0	0
$5$	1.65769	0.741344	0.370672	−	0.928764i	$-0.379127\pi$
$5$	1.65769	0.741344	0.370672	+	0.928764i	$0.379127\pi$
$6$	0	0
$7$	−0.109887	−0.0415333	−0.0207666	−	0.999784i	$-0.506611\pi$
$7$	−0.109887	−0.0415333	−0.0207666	+	0.999784i	$0.506611\pi$
$8$	0	0
$9$	1.19506	0.398352
$10$	0	0
$11$	1.18302	0.356693	0.178346	−	0.983968i	$-0.442925\pi$
$11$	1.18302	0.356693	0.178346	+	0.983968i	$0.442925\pi$
$12$	0	0
$13$	0.0953931	0.0264573	0.0132286	−	0.999912i	$-0.495789\pi$
$13$	0.0953931	0.0264573	0.0132286	+	0.999912i	$0.495789\pi$
$14$	0	0
$15$	−3.39526	−0.876653
$16$	0	0
$17$	4.88654	1.18516	0.592580	−	0.805511i	$-0.298109\pi$
$17$	4.88654	1.18516	0.592580	+	0.805511i	$0.298109\pi$
$18$	0	0
$19$	8.36455	1.91896	0.959480	−	0.281777i	$-0.0909239\pi$
$19$	8.36455	1.91896	0.959480	+	0.281777i	$0.0909239\pi$
$20$	0	0
$21$	0.225068	0.0491139
$22$	0	0
$23$	−8.76002	−1.82659	−0.913296	−	0.407297i	$-0.866471\pi$
$23$	−8.76002	−1.82659	−0.913296	+	0.407297i	$0.866471\pi$
$24$	0	0
$25$	−2.25205	−0.450410
$26$	0	0
$27$	3.69686	0.711460
$28$	0	0
$29$	6.50912	1.20871	0.604357	−	0.796714i	$-0.293430\pi$
$29$	6.50912	1.20871	0.604357	+	0.796714i	$0.293430\pi$
$30$	0	0
$31$	3.01815	0.542075	0.271038	−	0.962569i	$-0.412633\pi$
$31$	3.01815	0.542075	0.271038	+	0.962569i	$0.412633\pi$
$32$	0	0
$33$	−2.42303	−0.421796
$34$	0	0
$35$	−0.182159	−0.0307904
$36$	0	0
$37$	1.27668	0.209885	0.104943	−	0.994478i	$-0.466534\pi$
$37$	1.27668	0.209885	0.104943	+	0.994478i	$0.466534\pi$
$38$	0	0
$39$	−0.195383	−0.0312863
$40$	0	0
$41$	−10.4931	−1.63875	−0.819375	−	0.573258i	$-0.805679\pi$
$41$	−10.4931	−1.63875	−0.819375	+	0.573258i	$0.805679\pi$
$42$	0	0
$43$	9.50239	1.44910	0.724551	−	0.689221i	$-0.242047\pi$
$43$	9.50239	1.44910	0.724551	+	0.689221i	$0.242047\pi$
$44$	0	0
$45$	1.98104	0.295316
$46$	0	0
$47$	10.6601	1.55494	0.777471	−	0.628919i	$-0.216502\pi$
$47$	10.6601	1.55494	0.777471	+	0.628919i	$0.216502\pi$
$48$	0	0
$49$	−6.98792	−0.998275
$50$	0	0
$51$	−10.0085	−1.40148
$52$	0	0
$53$	−3.94211	−0.541490	−0.270745	−	0.962651i	$-0.587270\pi$
$53$	−3.94211	−0.541490	−0.270745	+	0.962651i	$0.587270\pi$
$54$	0	0
$55$	1.96108	0.264432
$56$	0	0
$57$	−17.1321	−2.26921
$58$	0	0
$59$	−4.33550	−0.564435	−0.282217	−	0.959351i	$-0.591070\pi$
$59$	−4.33550	−0.564435	−0.282217	+	0.959351i	$0.591070\pi$
$60$	0	0
$61$	7.35520	0.941737	0.470869	−	0.882203i	$-0.343941\pi$
$61$	7.35520	0.941737	0.470869	+	0.882203i	$0.343941\pi$
$62$	0	0
$63$	−0.131321	−0.0165449
$64$	0	0
$65$	0.158133	0.0196139
$66$	0	0
$67$	9.54182	1.16572	0.582859	−	0.812573i	$-0.301934\pi$
$67$	9.54182	1.16572	0.582859	+	0.812573i	$0.301934\pi$
$68$	0	0
$69$	17.9421	2.15998
$70$	0	0
$71$	2.52756	0.299966	0.149983	−	0.988689i	$-0.452078\pi$
$71$	2.52756	0.299966	0.149983	+	0.988689i	$0.452078\pi$
$72$	0	0
$73$	11.4899	1.34479	0.672397	−	0.740190i	$-0.265265\pi$
$73$	11.4899	1.34479	0.672397	+	0.740190i	$0.265265\pi$
$74$	0	0
$75$	4.61261	0.532618
$76$	0	0
$77$	−0.129998	−0.0148146
$78$	0	0
$79$	0.886677	0.0997589	0.0498795	−	0.998755i	$-0.484116\pi$
$79$	0.886677	0.0997589	0.0498795	+	0.998755i	$0.484116\pi$
$80$	0	0
$81$	−11.1570	−1.23967
$82$	0	0
$83$	−14.4973	−1.59128	−0.795641	−	0.605769i	$-0.792866\pi$
$83$	−14.4973	−1.59128	−0.795641	+	0.605769i	$0.792866\pi$
$84$	0	0
$85$	8.10039	0.878611
$86$	0	0
$87$	−13.3319	−1.42933
$88$	0	0
$89$	−7.41900	−0.786413	−0.393206	−	0.919450i	$-0.628634\pi$
$89$	−7.41900	−0.786413	−0.393206	+	0.919450i	$0.628634\pi$
$90$	0	0
$91$	−0.0104824	−0.00109886
$92$	0	0
$93$	−6.18172	−0.641014
$94$	0	0
$95$	13.8659	1.42261
$96$	0	0
$97$	−11.4811	−1.16573	−0.582865	−	0.812569i	$-0.698068\pi$
$97$	−11.4811	−1.16573	−0.582865	+	0.812569i	$0.698068\pi$
$98$	0	0
$99$	1.41377	0.142089
$100$	0	0
$101$	−7.89213	−0.785296	−0.392648	−	0.919689i	$-0.628441\pi$
$101$	−7.89213	−0.785296	−0.392648	+	0.919689i	$0.628441\pi$
$102$	0	0
$103$	2.03892	0.200901	0.100450	−	0.994942i	$-0.467972\pi$
$103$	2.03892	0.200901	0.100450	+	0.994942i	$0.467972\pi$
$104$	0	0
$105$	0.373094	0.0364103
$106$	0	0
$107$	−12.4480	−1.20340	−0.601698	−	0.798724i	$-0.705509\pi$
$107$	−12.4480	−1.20340	−0.601698	+	0.798724i	$0.705509\pi$
$108$	0	0
$109$	17.3506	1.66189	0.830943	−	0.556357i	$-0.187801\pi$
$109$	17.3506	1.66189	0.830943	+	0.556357i	$0.187801\pi$
$110$	0	0
$111$	−2.61488	−0.248193
$112$	0	0
$113$	5.71505	0.537627	0.268813	−	0.963192i	$-0.413369\pi$
$113$	5.71505	0.537627	0.268813	+	0.963192i	$0.413369\pi$
$114$	0	0
$115$	−14.5214	−1.35413
$116$	0	0
$117$	0.114000	0.0105393
$118$	0	0
$119$	−0.536966	−0.0492236
$120$	0	0
$121$	−9.60047	−0.872770
$122$	0	0
$123$	21.4918	1.93785
$124$	0	0
$125$	−12.0217	−1.07525
$126$	0	0
$127$	2.02599	0.179777	0.0898887	−	0.995952i	$-0.471349\pi$
$127$	2.02599	0.179777	0.0898887	+	0.995952i	$0.471349\pi$
$128$	0	0
$129$	−19.4626	−1.71359
$130$	0	0
$131$	19.5457	1.70772	0.853858	−	0.520506i	$-0.174257\pi$
$131$	19.5457	1.70772	0.853858	+	0.520506i	$0.174257\pi$
$132$	0	0
$133$	−0.919153	−0.0797007
$134$	0	0
$135$	6.12826	0.527437
$136$	0	0
$137$	−12.3692	−1.05677	−0.528387	−	0.849004i	$-0.677203\pi$
$137$	−12.3692	−1.05677	−0.528387	+	0.849004i	$0.677203\pi$
$138$	0	0
$139$	0.954207	0.0809348	0.0404674	−	0.999181i	$-0.487115\pi$
$139$	0.954207	0.0809348	0.0404674	+	0.999181i	$0.487115\pi$
$140$	0	0
$141$	−21.8339	−1.83875
$142$	0	0
$143$	0.112852	0.00943712
$144$	0	0
$145$	10.7901	0.896072
$146$	0	0
$147$	14.3126	1.18048
$148$	0	0
$149$	20.7400	1.69908	0.849542	−	0.527521i	$-0.176878\pi$
$149$	20.7400	1.69908	0.849542	+	0.527521i	$0.176878\pi$
$150$	0	0
$151$	−13.5948	−1.10633	−0.553163	−	0.833073i	$-0.686579\pi$
$151$	−13.5948	−1.10633	−0.553163	+	0.833073i	$0.686579\pi$
$152$	0	0
$153$	5.83969	0.472111
$154$	0	0
$155$	5.00317	0.401864
$156$	0	0
$157$	11.0815	0.884404	0.442202	−	0.896916i	$-0.354197\pi$
$157$	11.0815	0.884404	0.442202	+	0.896916i	$0.354197\pi$
$158$	0	0
$159$	8.07416	0.640322
$160$	0	0
$161$	0.962610	0.0758643
$162$	0	0
$163$	11.3018	0.885228	0.442614	−	0.896712i	$-0.354051\pi$
$163$	11.3018	0.885228	0.442614	+	0.896712i	$0.354051\pi$
$164$	0	0
$165$	−4.01665	−0.312696
$166$	0	0
$167$	25.6987	1.98862	0.994312	−	0.106502i	$-0.0339652\pi$
$167$	25.6987	1.98862	0.994312	+	0.106502i	$0.0339652\pi$
$168$	0	0
$169$	−12.9909	−0.999300
$170$	0	0
$171$	9.99611	0.764421
$172$	0	0
$173$	−12.6243	−0.959810	−0.479905	−	0.877321i	$-0.659329\pi$
$173$	−12.6243	−0.959810	−0.479905	+	0.877321i	$0.659329\pi$
$174$	0	0
$175$	0.247470	0.0187070
$176$	0	0
$177$	8.87991	0.667455
$178$	0	0
$179$	17.4406	1.30357	0.651786	−	0.758403i	$-0.274020\pi$
$179$	17.4406	1.30357	0.651786	+	0.758403i	$0.274020\pi$
$180$	0	0
$181$	−11.9219	−0.886149	−0.443075	−	0.896485i	$-0.646112\pi$
$181$	−11.9219	−0.886149	−0.443075	+	0.896485i	$0.646112\pi$
$182$	0	0
$183$	−15.0648	−1.11362
$184$	0	0
$185$	2.11635	0.155597
$186$	0	0
$187$	5.78085	0.422738
$188$	0	0
$189$	−0.406236	−0.0295493
$190$	0	0
$191$	−1.67995	−0.121557	−0.0607786	−	0.998151i	$-0.519358\pi$
$191$	−1.67995	−0.121557	−0.0607786	+	0.998151i	$0.519358\pi$
$192$	0	0
$193$	−1.63158	−0.117443	−0.0587217	−	0.998274i	$-0.518702\pi$
$193$	−1.63158	−0.117443	−0.0587217	+	0.998274i	$0.518702\pi$
$194$	0	0
$195$	−0.323885	−0.0231939
$196$	0	0
$197$	13.1605	0.937649	0.468825	−	0.883291i	$-0.344678\pi$
$197$	13.1605	0.937649	0.468825	+	0.883291i	$0.344678\pi$
$198$	0	0
$199$	−21.0149	−1.48971	−0.744853	−	0.667229i	$-0.767480\pi$
$199$	−21.0149	−1.48971	−0.744853	+	0.667229i	$0.767480\pi$
$200$	0	0
$201$	−19.5434	−1.37849
$202$	0	0
$203$	−0.715266	−0.0502018
$204$	0	0
$205$	−17.3944	−1.21488
$206$	0	0
$207$	−10.4687	−0.727626
$208$	0	0
$209$	9.89539	0.684479
$210$	0	0
$211$	18.8517	1.29781	0.648903	−	0.760871i	$-0.275228\pi$
$211$	18.8517	1.29781	0.648903	+	0.760871i	$0.275228\pi$
$212$	0	0
$213$	−5.17690	−0.354716
$214$	0	0
$215$	15.7521	1.07428
$216$	0	0
$217$	−0.331654	−0.0225142
$218$	0	0
$219$	−23.5335	−1.59025
$220$	0	0
$221$	0.466143	0.0313561
$222$	0	0
$223$	13.5477	0.907219	0.453610	−	0.891201i	$-0.350136\pi$
$223$	13.5477	0.907219	0.453610	+	0.891201i	$0.350136\pi$
$224$	0	0
$225$	−2.69132	−0.179422
$226$	0	0
$227$	−3.11841	−0.206976	−0.103488	−	0.994631i	$-0.533000\pi$
$227$	−3.11841	−0.206976	−0.103488	+	0.994631i	$0.533000\pi$
$228$	0	0
$229$	−2.47507	−0.163557	−0.0817786	−	0.996651i	$-0.526060\pi$
$229$	−2.47507	−0.163557	−0.0817786	+	0.996651i	$0.526060\pi$
$230$	0	0
$231$	0.266259	0.0175186
$232$	0	0
$233$	−21.0542	−1.37931	−0.689655	−	0.724138i	$-0.742238\pi$
$233$	−21.0542	−1.37931	−0.689655	+	0.724138i	$0.742238\pi$
$234$	0	0
$235$	17.6713	1.15275
$236$	0	0
$237$	−1.81608	−0.117967
$238$	0	0
$239$	23.4878	1.51930	0.759651	−	0.650331i	$-0.225370\pi$
$239$	23.4878	1.51930	0.759651	+	0.650331i	$0.225370\pi$
$240$	0	0
$241$	−8.41632	−0.542143	−0.271071	−	0.962559i	$-0.587378\pi$
$241$	−8.41632	−0.542143	−0.271071	+	0.962559i	$0.587378\pi$
$242$	0	0
$243$	11.7610	0.754470
$244$	0	0
$245$	−11.5838	−0.740065
$246$	0	0
$247$	0.797921	0.0507705
$248$	0	0
$249$	29.6931	1.88172
$250$	0	0
$251$	−18.8550	−1.19012	−0.595058	−	0.803683i	$-0.702871\pi$
$251$	−18.8550	−1.19012	−0.595058	+	0.803683i	$0.702871\pi$
$252$	0	0
$253$	−10.3632	−0.651532
$254$	0	0
$255$	−16.5911	−1.03897
$256$	0	0
$257$	16.7747	1.04638	0.523188	−	0.852217i	$-0.324743\pi$
$257$	16.7747	1.04638	0.523188	+	0.852217i	$0.324743\pi$
$258$	0	0
$259$	−0.140290	−0.00871722
$260$	0	0
$261$	7.77876	0.481493
$262$	0	0
$263$	18.2741	1.12683	0.563416	−	0.826173i	$-0.309487\pi$
$263$	18.2741	1.12683	0.563416	+	0.826173i	$0.309487\pi$
$264$	0	0
$265$	−6.53481	−0.401430
$266$	0	0
$267$	15.1955	0.929948
$268$	0	0
$269$	−28.6373	−1.74605	−0.873024	−	0.487677i	$-0.837844\pi$
$269$	−28.6373	−1.74605	−0.873024	+	0.487677i	$0.837844\pi$
$270$	0	0
$271$	−18.9893	−1.15351	−0.576757	−	0.816915i	$-0.695682\pi$
$271$	−18.9893	−1.15351	−0.576757	+	0.816915i	$0.695682\pi$
$272$	0	0
$273$	0.0214700	0.00129942
$274$	0	0
$275$	−2.66421	−0.160658
$276$	0	0
$277$	12.3030	0.739216	0.369608	−	0.929188i	$-0.379492\pi$
$277$	12.3030	0.739216	0.369608	+	0.929188i	$0.379492\pi$
$278$	0	0
$279$	3.60685	0.215937
$280$	0	0
$281$	−25.9601	−1.54865	−0.774324	−	0.632789i	$-0.781910\pi$
$281$	−25.9601	−1.54865	−0.774324	+	0.632789i	$0.781910\pi$
$282$	0	0
$283$	9.74437	0.579243	0.289621	−	0.957141i	$-0.406471\pi$
$283$	9.74437	0.579243	0.289621	+	0.957141i	$0.406471\pi$
$284$	0	0
$285$	−28.3999	−1.68226
$286$	0	0
$287$	1.15305	0.0680627
$288$	0	0
$289$	6.87830	0.404606
$290$	0	0
$291$	23.5154	1.37850
$292$	0	0
$293$	25.1947	1.47189	0.735944	−	0.677042i	$-0.236739\pi$
$293$	25.1947	1.47189	0.735944	+	0.677042i	$0.236739\pi$
$294$	0	0
$295$	−7.18694	−0.418440
$296$	0	0
$297$	4.37344	0.253773
$298$	0	0
$299$	−0.835646	−0.0483267
$300$	0	0
$301$	−1.04419	−0.0601859
$302$	0	0
$303$	16.1645	0.928628
$304$	0	0
$305$	12.1927	0.698151
$306$	0	0
$307$	−29.7625	−1.69864	−0.849319	−	0.527881i	$-0.822987\pi$
$307$	−29.7625	−1.69864	−0.849319	+	0.527881i	$0.822987\pi$
$308$	0	0
$309$	−4.17608	−0.237569
$310$	0	0
$311$	−11.2171	−0.636064	−0.318032	−	0.948080i	$-0.603022\pi$
$311$	−11.2171	−0.636064	−0.318032	+	0.948080i	$0.603022\pi$
$312$	0	0
$313$	−0.330383	−0.0186744	−0.00933718	−	0.999956i	$-0.502972\pi$
$313$	−0.330383	−0.0186744	−0.00933718	+	0.999956i	$0.502972\pi$
$314$	0	0
$315$	−0.217690	−0.0122654
$316$	0	0
$317$	23.9307	1.34408	0.672040	−	0.740515i	$-0.265418\pi$
$317$	23.9307	1.34408	0.672040	+	0.740515i	$0.265418\pi$
$318$	0	0
$319$	7.70039	0.431139
$320$	0	0
$321$	25.4958	1.42304
$322$	0	0
$323$	40.8737	2.27428
$324$	0	0
$325$	−0.214830	−0.0119166
$326$	0	0
$327$	−35.5372	−1.96521
$328$	0	0
$329$	−1.17141	−0.0645818
$330$	0	0
$331$	20.5129	1.12749	0.563746	−	0.825948i	$-0.309360\pi$
$331$	20.5129	1.12749	0.563746	+	0.825948i	$0.309360\pi$
$332$	0	0
$333$	1.52571	0.0836082
$334$	0	0
$335$	15.8174	0.864198
$336$	0	0
$337$	−16.9971	−0.925889	−0.462945	−	0.886387i	$-0.653207\pi$
$337$	−16.9971	−0.925889	−0.462945	+	0.886387i	$0.653207\pi$
$338$	0	0
$339$	−11.7055	−0.635754
$340$	0	0
$341$	3.57051	0.193354
$342$	0	0
$343$	1.53709	0.0829949
$344$	0	0
$345$	29.7426	1.60129
$346$	0	0
$347$	−16.6417	−0.893374	−0.446687	−	0.894690i	$-0.647396\pi$
$347$	−16.6417	−0.893374	−0.446687	+	0.894690i	$0.647396\pi$
$348$	0	0
$349$	33.9609	1.81789	0.908943	−	0.416921i	$-0.136891\pi$
$349$	33.9609	1.81789	0.908943	+	0.416921i	$0.136891\pi$
$350$	0	0
$351$	0.352655	0.0188233
$352$	0	0
$353$	25.2721	1.34510	0.672550	−	0.740052i	$-0.265199\pi$
$353$	25.2721	1.34510	0.672550	+	0.740052i	$0.265199\pi$
$354$	0	0
$355$	4.18992	0.222378
$356$	0	0
$357$	1.09981	0.0582079
$358$	0	0
$359$	−10.5439	−0.556487	−0.278244	−	0.960511i	$-0.589752\pi$
$359$	−10.5439	−0.556487	−0.278244	+	0.960511i	$0.589752\pi$
$360$	0	0
$361$	50.9657	2.68241
$362$	0	0
$363$	19.6635	1.03207
$364$	0	0
$365$	19.0468	0.996955
$366$	0	0
$367$	−1.50088	−0.0783454	−0.0391727	−	0.999232i	$-0.512472\pi$
$367$	−1.50088	−0.0783454	−0.0391727	+	0.999232i	$0.512472\pi$
$368$	0	0
$369$	−12.5399	−0.652799
$370$	0	0
$371$	0.433185	0.0224899
$372$	0	0
$373$	−13.5214	−0.700112	−0.350056	−	0.936729i	$-0.613838\pi$
$373$	−13.5214	−0.700112	−0.350056	+	0.936729i	$0.613838\pi$
$374$	0	0
$375$	24.6226	1.27151
$376$	0	0
$377$	0.620925	0.0319793
$378$	0	0
$379$	20.2095	1.03809	0.519045	−	0.854747i	$-0.326288\pi$
$379$	20.2095	1.03809	0.519045	+	0.854747i	$0.326288\pi$
$380$	0	0
$381$	−4.14959	−0.212590
$382$	0	0
$383$	−17.2087	−0.879322	−0.439661	−	0.898164i	$-0.644901\pi$
$383$	−17.2087	−0.879322	−0.439661	+	0.898164i	$0.644901\pi$
$384$	0	0
$385$	−0.215496	−0.0109827
$386$	0	0
$387$	11.3559	0.577252
$388$	0	0
$389$	−20.7228	−1.05069	−0.525344	−	0.850890i	$-0.676064\pi$
$389$	−20.7228	−1.05069	−0.525344	+	0.850890i	$0.676064\pi$
$390$	0	0
$391$	−42.8062	−2.16480
$392$	0	0
$393$	−40.0332	−2.01941
$394$	0	0
$395$	1.46984	0.0739556
$396$	0	0
$397$	14.6264	0.734079	0.367040	−	0.930205i	$-0.380371\pi$
$397$	14.6264	0.734079	0.367040	+	0.930205i	$0.380371\pi$
$398$	0	0
$399$	1.88259	0.0942476
$400$	0	0
$401$	33.4644	1.67113	0.835565	−	0.549392i	$-0.185140\pi$
$401$	33.4644	1.67113	0.835565	+	0.549392i	$0.185140\pi$
$402$	0	0
$403$	0.287910	0.0143418
$404$	0	0
$405$	−18.4949	−0.919020
$406$	0	0
$407$	1.51033	0.0748645
$408$	0	0
$409$	13.5759	0.671285	0.335643	−	0.941989i	$-0.391047\pi$
$409$	13.5759	0.671285	0.335643	+	0.941989i	$0.391047\pi$
$410$	0	0
$411$	25.3344	1.24966
$412$	0	0
$413$	0.476414	0.0234428
$414$	0	0
$415$	−24.0320	−1.17969
$416$	0	0
$417$	−1.95439	−0.0957070
$418$	0	0
$419$	−6.04987	−0.295556	−0.147778	−	0.989021i	$-0.547212\pi$
$419$	−6.04987	−0.295556	−0.147778	+	0.989021i	$0.547212\pi$
$420$	0	0
$421$	−18.8175	−0.917110	−0.458555	−	0.888666i	$-0.651633\pi$
$421$	−18.8175	−0.917110	−0.458555	+	0.888666i	$0.651633\pi$
$422$	0	0
$423$	12.7395	0.619414
$424$	0	0
$425$	−11.0047	−0.533808
$426$	0	0
$427$	−0.808239	−0.0391134
$428$	0	0
$429$	−0.231141	−0.0111596
$430$	0	0
$431$	−12.5142	−0.602786	−0.301393	−	0.953500i	$-0.597452\pi$
$431$	−12.5142	−0.602786	−0.301393	+	0.953500i	$0.597452\pi$
$432$	0	0
$433$	29.5099	1.41815	0.709077	−	0.705131i	$-0.249112\pi$
$433$	29.5099	1.41815	0.709077	+	0.705131i	$0.249112\pi$
$434$	0	0
$435$	−22.1002	−1.05962
$436$	0	0
$437$	−73.2737	−3.50516
$438$	0	0
$439$	11.2346	0.536197	0.268098	−	0.963392i	$-0.413605\pi$
$439$	11.2346	0.536197	0.268098	+	0.963392i	$0.413605\pi$
$440$	0	0
$441$	−8.35096	−0.397665
$442$	0	0
$443$	19.9901	0.949761	0.474880	−	0.880050i	$-0.342491\pi$
$443$	19.9901	0.949761	0.474880	+	0.880050i	$0.342491\pi$
$444$	0	0
$445$	−12.2984	−0.583002
$446$	0	0
$447$	−42.4793	−2.00920
$448$	0	0
$449$	−20.7582	−0.979638	−0.489819	−	0.871824i	$-0.662937\pi$
$449$	−20.7582	−0.979638	−0.489819	+	0.871824i	$0.662937\pi$
$450$	0	0
$451$	−12.4135	−0.584530
$452$	0	0
$453$	27.8445	1.30825
$454$	0	0
$455$	−0.0173767	−0.000814631 0
$456$	0	0
$457$	39.7854	1.86108	0.930540	−	0.366189i	$-0.119338\pi$
$457$	39.7854	1.86108	0.930540	+	0.366189i	$0.119338\pi$
$458$	0	0
$459$	18.0648	0.843195
$460$	0	0
$461$	−22.5825	−1.05177	−0.525886	−	0.850555i	$-0.676266\pi$
$461$	−22.5825	−1.05177	−0.525886	+	0.850555i	$0.676266\pi$
$462$	0	0
$463$	12.2282	0.568293	0.284146	−	0.958781i	$-0.408290\pi$
$463$	12.2282	0.568293	0.284146	+	0.958781i	$0.408290\pi$
$464$	0	0
$465$	−10.2474	−0.475212
$466$	0	0
$467$	23.3575	1.08086	0.540429	−	0.841390i	$-0.318262\pi$
$467$	23.3575	1.08086	0.540429	+	0.841390i	$0.318262\pi$
$468$	0	0
$469$	−1.04852	−0.0484161
$470$	0	0
$471$	−22.6970	−1.04582
$472$	0	0
$473$	11.2415	0.516884
$474$	0	0
$475$	−18.8374	−0.864318
$476$	0	0
$477$	−4.71104	−0.215704
$478$	0	0
$479$	19.0833	0.871940	0.435970	−	0.899961i	$-0.356405\pi$
$479$	19.0833	0.871940	0.435970	+	0.899961i	$0.356405\pi$
$480$	0	0
$481$	0.121787	0.00555300
$482$	0	0
$483$	−1.97160	−0.0897110
$484$	0	0
$485$	−19.0322	−0.864206
$486$	0	0
$487$	−14.4237	−0.653600	−0.326800	−	0.945093i	$-0.605970\pi$
$487$	−14.4237	−0.653600	−0.326800	+	0.945093i	$0.605970\pi$
$488$	0	0
$489$	−23.1482	−1.04680
$490$	0	0
$491$	3.52012	0.158861	0.0794304	−	0.996840i	$-0.474690\pi$
$491$	3.52012	0.158861	0.0794304	+	0.996840i	$0.474690\pi$
$492$	0	0
$493$	31.8071	1.43252
$494$	0	0
$495$	2.34360	0.105337
$496$	0	0
$497$	−0.277745	−0.0124586
$498$	0	0
$499$	37.1469	1.66292	0.831462	−	0.555581i	$-0.187504\pi$
$499$	37.1469	1.66292	0.831462	+	0.555581i	$0.187504\pi$
$500$	0	0
$501$	−52.6356	−2.35159
$502$	0	0
$503$	35.2409	1.57131	0.785656	−	0.618663i	$-0.212325\pi$
$503$	35.2409	1.57131	0.785656	+	0.618663i	$0.212325\pi$
$504$	0	0
$505$	−13.0827	−0.582174
$506$	0	0
$507$	26.6077	1.18169
$508$	0	0
$509$	35.2506	1.56245	0.781227	−	0.624247i	$-0.214594\pi$
$509$	35.2506	1.56245	0.781227	+	0.624247i	$0.214594\pi$
$510$	0	0
$511$	−1.26259	−0.0558537
$512$	0	0
$513$	30.9226	1.36526
$514$	0	0
$515$	3.37991	0.148937
$516$	0	0
$517$	12.6111	0.554636
$518$	0	0
$519$	25.8569	1.13499
$520$	0	0
$521$	−12.6483	−0.554133	−0.277067	−	0.960851i	$-0.589362\pi$
$521$	−12.6483	−0.554133	−0.277067	+	0.960851i	$0.589362\pi$
$522$	0	0
$523$	26.2438	1.14756	0.573780	−	0.819010i	$-0.305476\pi$
$523$	26.2438	1.14756	0.573780	+	0.819010i	$0.305476\pi$
$524$	0	0
$525$	−0.506865	−0.0221214
$526$	0	0
$527$	14.7483	0.642446
$528$	0	0
$529$	53.7380	2.33644
$530$	0	0
$531$	−5.18117	−0.224844
$532$	0	0
$533$	−1.00097	−0.0433569
$534$	0	0
$535$	−20.6350	−0.892130
$536$	0	0
$537$	−35.7216	−1.54150
$538$	0	0
$539$	−8.26682	−0.356077
$540$	0	0
$541$	−26.5972	−1.14350	−0.571750	−	0.820428i	$-0.693735\pi$
$541$	−26.5972	−1.14350	−0.571750	+	0.820428i	$0.693735\pi$
$542$	0	0
$543$	24.4183	1.04789
$544$	0	0
$545$	28.7620	1.23203
$546$	0	0
$547$	3.71954	0.159036	0.0795180	−	0.996833i	$-0.474662\pi$
$547$	3.71954	0.159036	0.0795180	+	0.996833i	$0.474662\pi$
$548$	0	0
$549$	8.78988	0.375143
$550$	0	0
$551$	54.4459	2.31947
$552$	0	0
$553$	−0.0974340	−0.00414332
$554$	0	0
$555$	−4.33467	−0.183997
$556$	0	0
$557$	−26.3375	−1.11596	−0.557979	−	0.829855i	$-0.688423\pi$
$557$	−26.3375	−1.11596	−0.557979	+	0.829855i	$0.688423\pi$
$558$	0	0
$559$	0.906463	0.0383393
$560$	0	0
$561$	−11.8403	−0.499896
$562$	0	0
$563$	14.4142	0.607484	0.303742	−	0.952754i	$-0.401764\pi$
$563$	14.4142	0.607484	0.303742	+	0.952754i	$0.401764\pi$
$564$	0	0
$565$	9.47381	0.398566
$566$	0	0
$567$	1.22601	0.0514875
$568$	0	0
$569$	4.37223	0.183294	0.0916468	−	0.995792i	$-0.470787\pi$
$569$	4.37223	0.183294	0.0916468	+	0.995792i	$0.470787\pi$
$570$	0	0
$571$	23.8430	0.997800	0.498900	−	0.866659i	$-0.333737\pi$
$571$	23.8430	0.997800	0.498900	+	0.866659i	$0.333737\pi$
$572$	0	0
$573$	3.44085	0.143744
$574$	0	0
$575$	19.7280	0.822715
$576$	0	0
$577$	14.6536	0.610039	0.305019	−	0.952346i	$-0.401337\pi$
$577$	14.6536	0.610039	0.305019	+	0.952346i	$0.401337\pi$
$578$	0	0
$579$	3.34177	0.138879
$580$	0	0
$581$	1.59306	0.0660911
$582$	0	0
$583$	−4.66357	−0.193145
$584$	0	0
$585$	0.188977	0.00781325
$586$	0	0
$587$	35.7300	1.47473	0.737367	−	0.675492i	$-0.236069\pi$
$587$	35.7300	1.47473	0.737367	+	0.675492i	$0.236069\pi$
$588$	0	0
$589$	25.2454	1.04022
$590$	0	0
$591$	−26.9552	−1.10879
$592$	0	0
$593$	−17.5326	−0.719976	−0.359988	−	0.932957i	$-0.617219\pi$
$593$	−17.5326	−0.719976	−0.359988	+	0.932957i	$0.617219\pi$
$594$	0	0
$595$	−0.890126	−0.0364916
$596$	0	0
$597$	43.0423	1.76161
$598$	0	0
$599$	26.2361	1.07198	0.535988	−	0.844225i	$-0.319939\pi$
$599$	26.2361	1.07198	0.535988	+	0.844225i	$0.319939\pi$
$600$	0	0
$601$	7.35392	0.299973	0.149986	−	0.988688i	$-0.452077\pi$
$601$	7.35392	0.299973	0.149986	+	0.988688i	$0.452077\pi$
$602$	0	0
$603$	11.4030	0.464366
$604$	0	0
$605$	−15.9147	−0.647023
$606$	0	0
$607$	−25.6326	−1.04040	−0.520198	−	0.854046i	$-0.674142\pi$
$607$	−25.6326	−1.04040	−0.520198	+	0.854046i	$0.674142\pi$
$608$	0	0
$609$	1.46500	0.0593646
$610$	0	0
$611$	1.01690	0.0411396
$612$	0	0
$613$	29.9627	1.21018	0.605092	−	0.796156i	$-0.293136\pi$
$613$	29.9627	1.21018	0.605092	+	0.796156i	$0.293136\pi$
$614$	0	0
$615$	35.6269	1.43662
$616$	0	0
$617$	34.6377	1.39446	0.697231	−	0.716847i	$-0.254415\pi$
$617$	34.6377	1.39446	0.697231	+	0.716847i	$0.254415\pi$
$618$	0	0
$619$	9.01958	0.362528	0.181264	−	0.983435i	$-0.441981\pi$
$619$	9.01958	0.362528	0.181264	+	0.983435i	$0.441981\pi$
$620$	0	0
$621$	−32.3846	−1.29955
$622$	0	0
$623$	0.815250	0.0326623
$624$	0	0
$625$	−8.66803	−0.346721
$626$	0	0
$627$	−20.2676	−0.809409
$628$	0	0
$629$	6.23856	0.248748
$630$	0	0
$631$	9.62686	0.383239	0.191620	−	0.981469i	$-0.438626\pi$
$631$	9.62686	0.383239	0.191620	+	0.981469i	$0.438626\pi$
$632$	0	0
$633$	−38.6118	−1.53468
$634$	0	0
$635$	3.35847	0.133277
$636$	0	0
$637$	−0.666600	−0.0264117
$638$	0	0
$639$	3.02057	0.119492
$640$	0	0
$641$	5.61773	0.221887	0.110944	−	0.993827i	$-0.464613\pi$
$641$	5.61773	0.221887	0.110944	+	0.993827i	$0.464613\pi$
$642$	0	0
$643$	−16.8511	−0.664543	−0.332271	−	0.943184i	$-0.607815\pi$
$643$	−16.8511	−0.664543	−0.332271	+	0.943184i	$0.607815\pi$
$644$	0	0
$645$	−32.2631	−1.27036
$646$	0	0
$647$	5.27321	0.207311	0.103656	−	0.994613i	$-0.466946\pi$
$647$	5.27321	0.207311	0.103656	+	0.994613i	$0.466946\pi$
$648$	0	0
$649$	−5.12897	−0.201330
$650$	0	0
$651$	0.679289	0.0266234
$652$	0	0
$653$	−10.4598	−0.409324	−0.204662	−	0.978833i	$-0.565609\pi$
$653$	−10.4598	−0.409324	−0.204662	+	0.978833i	$0.565609\pi$
$654$	0	0
$655$	32.4008	1.26600
$656$	0	0
$657$	13.7311	0.535702
$658$	0	0
$659$	0.323478	0.0126009	0.00630045	−	0.999980i	$-0.497994\pi$
$659$	0.323478	0.0126009	0.00630045	+	0.999980i	$0.497994\pi$
$660$	0	0
$661$	36.7200	1.42824	0.714121	−	0.700022i	$-0.246827\pi$
$661$	36.7200	1.42824	0.714121	+	0.700022i	$0.246827\pi$
$662$	0	0
$663$	−0.954746	−0.0370792
$664$	0	0
$665$	−1.52368	−0.0590856
$666$	0	0
$667$	−57.0200	−2.20783
$668$	0	0
$669$	−27.7481	−1.07280
$670$	0	0
$671$	8.70132	0.335911
$672$	0	0
$673$	−21.9985	−0.847980	−0.423990	−	0.905667i	$-0.639371\pi$
$673$	−21.9985	−0.847980	−0.423990	+	0.905667i	$0.639371\pi$
$674$	0	0
$675$	−8.32550	−0.320449
$676$	0	0
$677$	15.0054	0.576702	0.288351	−	0.957525i	$-0.406893\pi$
$677$	15.0054	0.576702	0.288351	+	0.957525i	$0.406893\pi$
$678$	0	0
$679$	1.26162	0.0484166
$680$	0	0
$681$	6.38708	0.244753
$682$	0	0
$683$	32.4105	1.24015	0.620076	−	0.784542i	$-0.287102\pi$
$683$	32.4105	1.24015	0.620076	+	0.784542i	$0.287102\pi$
$684$	0	0
$685$	−20.5044	−0.783433
$686$	0	0
$687$	5.06940	0.193410
$688$	0	0
$689$	−0.376050	−0.0143264
$690$	0	0
$691$	−5.94540	−0.226174	−0.113087	−	0.993585i	$-0.536074\pi$
$691$	−5.94540	−0.226174	−0.113087	+	0.993585i	$0.536074\pi$
$692$	0	0
$693$	−0.155354	−0.00590143
$694$	0	0
$695$	1.58178	0.0600005
$696$	0	0
$697$	−51.2751	−1.94218
$698$	0	0
$699$	43.1230	1.63106
$700$	0	0
$701$	26.5657	1.00337	0.501686	−	0.865050i	$-0.332713\pi$
$701$	26.5657	1.00337	0.501686	+	0.865050i	$0.332713\pi$
$702$	0	0
$703$	10.6789	0.402761
$704$	0	0
$705$	−36.1940	−1.36314
$706$	0	0
$707$	0.867240	0.0326159
$708$	0	0
$709$	14.8390	0.557291	0.278646	−	0.960394i	$-0.410114\pi$
$709$	14.8390	0.557291	0.278646	+	0.960394i	$0.410114\pi$
$710$	0	0
$711$	1.05963	0.0397392
$712$	0	0
$713$	−26.4390	−0.990150
$714$	0	0
$715$	0.187073	0.00699615
$716$	0	0
$717$	−48.1074	−1.79660
$718$	0	0
$719$	−39.6664	−1.47931	−0.739653	−	0.672988i	$-0.765010\pi$
$719$	−39.6664	−1.47931	−0.739653	+	0.672988i	$0.765010\pi$
$720$	0	0
$721$	−0.224050	−0.00834407
$722$	0	0
$723$	17.2382	0.641094
$724$	0	0
$725$	−14.6589	−0.544416
$726$	0	0
$727$	32.7755	1.21558	0.607788	−	0.794099i	$-0.292057\pi$
$727$	32.7755	1.21558	0.607788	+	0.794099i	$0.292057\pi$
$728$	0	0
$729$	9.38228	0.347492
$730$	0	0
$731$	46.4339	1.71742
$732$	0	0
$733$	−8.92433	−0.329628	−0.164814	−	0.986325i	$-0.552702\pi$
$733$	−8.92433	−0.329628	−0.164814	+	0.986325i	$0.552702\pi$
$734$	0	0
$735$	23.7258	0.875141
$736$	0	0
$737$	11.2881	0.415803
$738$	0	0
$739$	−16.2070	−0.596185	−0.298092	−	0.954537i	$-0.596350\pi$
$739$	−16.2070	−0.596185	−0.298092	+	0.954537i	$0.596350\pi$
$740$	0	0
$741$	−1.63429	−0.0600371
$742$	0	0
$743$	40.2972	1.47836	0.739181	−	0.673507i	$-0.235213\pi$
$743$	40.2972	1.47836	0.739181	+	0.673507i	$0.235213\pi$
$744$	0	0
$745$	34.3805	1.25961
$746$	0	0
$747$	−17.3250	−0.633890
$748$	0	0
$749$	1.36787	0.0499810
$750$	0	0
$751$	13.2212	0.482450	0.241225	−	0.970469i	$-0.422451\pi$
$751$	13.2212	0.482450	0.241225	+	0.970469i	$0.422451\pi$
$752$	0	0
$753$	38.6185	1.40733
$754$	0	0
$755$	−22.5359	−0.820167
$756$	0	0
$757$	−18.4618	−0.671005	−0.335502	−	0.942039i	$-0.608906\pi$
$757$	−18.4618	−0.671005	−0.335502	+	0.942039i	$0.608906\pi$
$758$	0	0
$759$	21.2258	0.770449
$760$	0	0
$761$	19.2010	0.696036	0.348018	−	0.937488i	$-0.386855\pi$
$761$	19.2010	0.696036	0.348018	+	0.937488i	$0.386855\pi$
$762$	0	0
$763$	−1.90660	−0.0690236
$764$	0	0
$765$	9.68042	0.349996
$766$	0	0
$767$	−0.413577	−0.0149334
$768$	0	0
$769$	−1.77119	−0.0638707	−0.0319353	−	0.999490i	$-0.510167\pi$
$769$	−1.77119	−0.0638707	−0.0319353	+	0.999490i	$0.510167\pi$
$770$	0	0
$771$	−34.3576	−1.23736
$772$	0	0
$773$	−44.2088	−1.59008	−0.795039	−	0.606558i	$-0.792550\pi$
$773$	−44.2088	−1.59008	−0.795039	+	0.606558i	$0.792550\pi$
$774$	0	0
$775$	−6.79701	−0.244156
$776$	0	0
$777$	0.287341	0.0103083
$778$	0	0
$779$	−87.7702	−3.14470
$780$	0	0
$781$	2.99014	0.106996
$782$	0	0
$783$	24.0633	0.859952
$784$	0	0
$785$	18.3698	0.655647
$786$	0	0
$787$	−24.6948	−0.880276	−0.440138	−	0.897930i	$-0.645071\pi$
$787$	−24.6948	−0.880276	−0.440138	+	0.897930i	$0.645071\pi$
$788$	0	0
$789$	−37.4288	−1.33250
$790$	0	0
$791$	−0.628008	−0.0223294
$792$	0	0
$793$	0.701636	0.0249158
$794$	0	0
$795$	13.3845	0.474699
$796$	0	0
$797$	43.3182	1.53441	0.767204	−	0.641403i	$-0.221647\pi$
$797$	43.3182	1.53441	0.767204	+	0.641403i	$0.221647\pi$
$798$	0	0
$799$	52.0912	1.84286
$800$	0	0
$801$	−8.86612	−0.313269
$802$	0	0
$803$	13.5928	0.479678
$804$	0	0
$805$	1.59571	0.0562415
$806$	0	0
$807$	58.6545	2.06474
$808$	0	0
$809$	−3.48956	−0.122687	−0.0613433	−	0.998117i	$-0.519538\pi$
$809$	−3.48956	−0.122687	−0.0613433	+	0.998117i	$0.519538\pi$
$810$	0	0
$811$	−17.8822	−0.627928	−0.313964	−	0.949435i	$-0.601657\pi$
$811$	−17.8822	−0.627928	−0.313964	+	0.949435i	$0.601657\pi$
$812$	0	0
$813$	38.8935	1.36405
$814$	0	0
$815$	18.7350	0.656258
$816$	0	0
$817$	79.4833	2.78077
$818$	0	0
$819$	−0.0125271	−0.000437732 0
$820$	0	0
$821$	−39.8068	−1.38927	−0.694633	−	0.719364i	$-0.744433\pi$
$821$	−39.8068	−1.38927	−0.694633	+	0.719364i	$0.744433\pi$
$822$	0	0
$823$	−18.3066	−0.638127	−0.319064	−	0.947733i	$-0.603368\pi$
$823$	−18.3066	−0.638127	−0.319064	+	0.947733i	$0.603368\pi$
$824$	0	0
$825$	5.45679	0.189981
$826$	0	0
$827$	27.9355	0.971412	0.485706	−	0.874122i	$-0.338563\pi$
$827$	27.9355	0.971412	0.485706	+	0.874122i	$0.338563\pi$
$828$	0	0
$829$	−31.6357	−1.09875	−0.549377	−	0.835575i	$-0.685135\pi$
$829$	−31.6357	−1.09875	−0.549377	+	0.835575i	$0.685135\pi$
$830$	0	0
$831$	−25.1988	−0.874137
$832$	0	0
$833$	−34.1468	−1.18312
$834$	0	0
$835$	42.6006	1.47425
$836$	0	0
$837$	11.1577	0.385665
$838$	0	0
$839$	−4.55934	−0.157406	−0.0787030	−	0.996898i	$-0.525078\pi$
$839$	−4.55934	−0.157406	−0.0787030	+	0.996898i	$0.525078\pi$
$840$	0	0
$841$	13.3686	0.460988
$842$	0	0
$843$	53.1710	1.83131
$844$	0	0
$845$	−21.5349	−0.740825
$846$	0	0
$847$	1.05496	0.0362490
$848$	0	0
$849$	−19.9583	−0.684966
$850$	0	0
$851$	−11.1838	−0.383375
$852$	0	0
$853$	24.2054	0.828778	0.414389	−	0.910100i	$-0.363995\pi$
$853$	24.2054	0.828778	0.414389	+	0.910100i	$0.363995\pi$
$854$	0	0
$855$	16.5705	0.566699
$856$	0	0
$857$	42.2122	1.44194	0.720971	−	0.692965i	$-0.243696\pi$
$857$	42.2122	1.44194	0.720971	+	0.692965i	$0.243696\pi$
$858$	0	0
$859$	−7.21927	−0.246318	−0.123159	−	0.992387i	$-0.539303\pi$
$859$	−7.21927	−0.246318	−0.123159	+	0.992387i	$0.539303\pi$
$860$	0	0
$861$	−2.36167	−0.0804854
$862$	0	0
$863$	10.0962	0.343679	0.171840	−	0.985125i	$-0.445029\pi$
$863$	10.0962	0.343679	0.171840	+	0.985125i	$0.445029\pi$
$864$	0	0
$865$	−20.9273	−0.711549
$866$	0	0
$867$	−14.0880	−0.478454
$868$	0	0
$869$	1.04895	0.0355833
$870$	0	0
$871$	0.910224	0.0308418
$872$	0	0
$873$	−13.7206	−0.464370
$874$	0	0
$875$	1.32102	0.0446587
$876$	0	0
$877$	−6.18782	−0.208948	−0.104474	−	0.994528i	$-0.533316\pi$
$877$	−6.18782	−0.208948	−0.104474	+	0.994528i	$0.533316\pi$
$878$	0	0
$879$	−51.6033	−1.74054
$880$	0	0
$881$	50.2851	1.69415	0.847074	−	0.531475i	$-0.178362\pi$
$881$	50.2851	1.69415	0.847074	+	0.531475i	$0.178362\pi$
$882$	0	0
$883$	2.55580	0.0860096	0.0430048	−	0.999075i	$-0.486307\pi$
$883$	2.55580	0.0860096	0.0430048	+	0.999075i	$0.486307\pi$
$884$	0	0
$885$	14.7202	0.494813
$886$	0	0
$887$	12.0900	0.405942	0.202971	−	0.979185i	$-0.434940\pi$
$887$	12.0900	0.405942	0.202971	+	0.979185i	$0.434940\pi$
$888$	0	0
$889$	−0.222629	−0.00746674
$890$	0	0
$891$	−13.1989	−0.442180
$892$	0	0
$893$	89.1673	2.98387
$894$	0	0
$895$	28.9112	0.966394
$896$	0	0
$897$	1.71156	0.0571472
$898$	0	0
$899$	19.6455	0.655213
$900$	0	0
$901$	−19.2633	−0.641753
$902$	0	0
$903$	2.13869	0.0711710
$904$	0	0
$905$	−19.7629	−0.656941
$906$	0	0
$907$	−1.66359	−0.0552388	−0.0276194	−	0.999619i	$-0.508793\pi$
$907$	−1.66359	−0.0552388	−0.0276194	+	0.999619i	$0.508793\pi$
$908$	0	0
$909$	−9.43154	−0.312824
$910$	0	0
$911$	−12.5133	−0.414584	−0.207292	−	0.978279i	$-0.566465\pi$
$911$	−12.5133	−0.414584	−0.207292	+	0.978279i	$0.566465\pi$
$912$	0	0
$913$	−17.1505	−0.567598
$914$	0	0
$915$	−24.9729	−0.825577
$916$	0	0
$917$	−2.14781	−0.0709270
$918$	0	0
$919$	−2.28682	−0.0754351	−0.0377176	−	0.999288i	$-0.512009\pi$
$919$	−2.28682	−0.0754351	−0.0377176	+	0.999288i	$0.512009\pi$
$920$	0	0
$921$	60.9591	2.00867
$922$	0	0
$923$	0.241112	0.00793629
$924$	0	0
$925$	−2.87515	−0.0945344
$926$	0	0
$927$	2.43662	0.0800292
$928$	0	0
$929$	−59.1346	−1.94014	−0.970072	−	0.242817i	$-0.921929\pi$
$929$	−59.1346	−1.94014	−0.970072	+	0.242817i	$0.921929\pi$
$930$	0	0
$931$	−58.4509	−1.91565
$932$	0	0
$933$	22.9747	0.752158
$934$	0	0
$935$	9.58289	0.313394
$936$	0	0
$937$	−21.4347	−0.700242	−0.350121	−	0.936704i	$-0.613860\pi$
$937$	−21.4347	−0.700242	−0.350121	+	0.936704i	$0.613860\pi$
$938$	0	0
$939$	0.676686	0.0220828
$940$	0	0
$941$	−41.6505	−1.35777	−0.678883	−	0.734246i	$-0.737536\pi$
$941$	−41.6505	−1.35777	−0.678883	+	0.734246i	$0.737536\pi$
$942$	0	0
$943$	91.9200	2.99333
$944$	0	0
$945$	−0.673414	−0.0219062
$946$	0	0
$947$	−3.42207	−0.111202	−0.0556012	−	0.998453i	$-0.517708\pi$
$947$	−3.42207	−0.111202	−0.0556012	+	0.998453i	$0.517708\pi$
$948$	0	0
$949$	1.09606	0.0355796
$950$	0	0
$951$	−49.0144	−1.58940
$952$	0	0
$953$	−54.3766	−1.76143	−0.880715	−	0.473647i	$-0.842937\pi$
$953$	−54.3766	−1.76143	−0.880715	+	0.473647i	$0.842937\pi$
$954$	0	0
$955$	−2.78485	−0.0901156
$956$	0	0
$957$	−15.7718	−0.509830
$958$	0	0
$959$	1.35921	0.0438913
$960$	0	0
$961$	−21.8908	−0.706155
$962$	0	0
$963$	−14.8761	−0.479375
$964$	0	0
$965$	−2.70465	−0.0870659
$966$	0	0
$967$	−41.4861	−1.33410	−0.667051	−	0.745012i	$-0.732444\pi$
$967$	−41.4861	−1.33410	−0.667051	+	0.745012i	$0.732444\pi$
$968$	0	0
$969$	−83.7169	−2.68937
$970$	0	0
$971$	−24.6011	−0.789487	−0.394743	−	0.918791i	$-0.629166\pi$
$971$	−24.6011	−0.789487	−0.394743	+	0.918791i	$0.629166\pi$
$972$	0	0
$973$	−0.104855	−0.00336149
$974$	0	0
$975$	0.440011	0.0140916
$976$	0	0
$977$	39.3715	1.25961	0.629803	−	0.776755i	$-0.283136\pi$
$977$	39.3715	1.25961	0.629803	+	0.776755i	$0.283136\pi$
$978$	0	0
$979$	−8.77679	−0.280508
$980$	0	0
$981$	20.7349	0.662016
$982$	0	0
$983$	−12.5999	−0.401874	−0.200937	−	0.979604i	$-0.564399\pi$
$983$	−12.5999	−0.401874	−0.200937	+	0.979604i	$0.564399\pi$
$984$	0	0
$985$	21.8161	0.695120
$986$	0	0
$987$	2.39926	0.0763693
$988$	0	0
$989$	−83.2412	−2.64692
$990$	0	0
$991$	18.2025	0.578221	0.289110	−	0.957296i	$-0.406641\pi$
$991$	18.2025	0.578221	0.289110	+	0.957296i	$0.406641\pi$
$992$	0	0
$993$	−42.0142	−1.33328
$994$	0	0
$995$	−34.8363	−1.10438
$996$	0	0
$997$	17.0611	0.540332	0.270166	−	0.962814i	$-0.412921\pi$
$997$	17.0611	0.540332	0.270166	+	0.962814i	$0.412921\pi$
$998$	0	0
$999$	4.71971	0.149325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8012.2.a.b.1.17	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8012.2.a.b.1.17	✓	88	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:	\( N \)	\(=\)	\( 8012 = 2^{2} \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.04818 q^{3} +1.65769 q^{5} -0.109887 q^{7} +1.19506 q^{9} +O(q^{10})\)	\(q-2.04818 q^{3} +1.65769 q^{5} -0.109887 q^{7} +1.19506 q^{9} +1.18302 q^{11} +0.0953931 q^{13} -3.39526 q^{15} +4.88654 q^{17} +8.36455 q^{19} +0.225068 q^{21} -8.76002 q^{23} -2.25205 q^{25} +3.69686 q^{27} +6.50912 q^{29} +3.01815 q^{31} -2.42303 q^{33} -0.182159 q^{35} +1.27668 q^{37} -0.195383 q^{39} -10.4931 q^{41} +9.50239 q^{43} +1.98104 q^{45} +10.6601 q^{47} -6.98792 q^{49} -10.0085 q^{51} -3.94211 q^{53} +1.96108 q^{55} -17.1321 q^{57} -4.33550 q^{59} +7.35520 q^{61} -0.131321 q^{63} +0.158133 q^{65} +9.54182 q^{67} +17.9421 q^{69} +2.52756 q^{71} +11.4899 q^{73} +4.61261 q^{75} -0.129998 q^{77} +0.886677 q^{79} -11.1570 q^{81} -14.4973 q^{83} +8.10039 q^{85} -13.3319 q^{87} -7.41900 q^{89} -0.0104824 q^{91} -6.18172 q^{93} +13.8659 q^{95} -11.4811 q^{97} +1.41377 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more