LMFDB - Embedded newform 8012.2.a.b.1.14

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			1.14
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)

$f(q)$	$=$	$q-2.24743 q^{3} -3.82088 q^{5} -1.51483 q^{7} +2.05096 q^{9} +O(q^{10})$	$q-2.24743 q^{3} -3.82088 q^{5} -1.51483 q^{7} +2.05096 q^{9} +4.42397 q^{11} +5.40932 q^{13} +8.58718 q^{15} +0.182540 q^{17} +5.74741 q^{19} +3.40449 q^{21} -2.79100 q^{23} +9.59912 q^{25} +2.13290 q^{27} +2.17337 q^{29} +4.26026 q^{31} -9.94257 q^{33} +5.78800 q^{35} +7.41092 q^{37} -12.1571 q^{39} +5.43554 q^{41} +7.88203 q^{43} -7.83648 q^{45} -8.32892 q^{47} -4.70528 q^{49} -0.410247 q^{51} -4.23320 q^{53} -16.9034 q^{55} -12.9169 q^{57} -0.259033 q^{59} +3.10398 q^{61} -3.10687 q^{63} -20.6684 q^{65} +9.04527 q^{67} +6.27258 q^{69} -5.83452 q^{71} +8.42397 q^{73} -21.5734 q^{75} -6.70157 q^{77} +0.253285 q^{79} -10.9464 q^{81} +3.29791 q^{83} -0.697464 q^{85} -4.88452 q^{87} +10.5409 q^{89} -8.19422 q^{91} -9.57466 q^{93} -21.9602 q^{95} +1.40835 q^{97} +9.07338 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.24743	−1.29756	−0.648778	−	0.760977i	$-0.724720\pi$
$3$	−2.24743	−1.29756	−0.648778	+	0.760977i	$0.724720\pi$
$4$	0	0
$5$	−3.82088	−1.70875	−0.854375	−	0.519658i	$-0.826060\pi$
$5$	−3.82088	−1.70875	−0.854375	+	0.519658i	$0.826060\pi$
$6$	0	0
$7$	−1.51483	−0.572553	−0.286277	−	0.958147i	$-0.592418\pi$
$7$	−1.51483	−0.572553	−0.286277	+	0.958147i	$0.592418\pi$
$8$	0	0
$9$	2.05096	0.683654
$10$	0	0
$11$	4.42397	1.33388	0.666938	−	0.745113i	$-0.267605\pi$
$11$	4.42397	1.33388	0.666938	+	0.745113i	$0.267605\pi$
$12$	0	0
$13$	5.40932	1.50028	0.750138	−	0.661281i	$-0.229987\pi$
$13$	5.40932	1.50028	0.750138	+	0.661281i	$0.229987\pi$
$14$	0	0
$15$	8.58718	2.21720
$16$	0	0
$17$	0.182540	0.0442725	0.0221362	−	0.999755i	$-0.492953\pi$
$17$	0.182540	0.0442725	0.0221362	+	0.999755i	$0.492953\pi$
$18$	0	0
$19$	5.74741	1.31855	0.659273	−	0.751903i	$-0.270864\pi$
$19$	5.74741	1.31855	0.659273	+	0.751903i	$0.270864\pi$
$20$	0	0
$21$	3.40449	0.742921
$22$	0	0
$23$	−2.79100	−0.581963	−0.290982	−	0.956729i	$-0.593982\pi$
$23$	−2.79100	−0.581963	−0.290982	+	0.956729i	$0.593982\pi$
$24$	0	0
$25$	9.59912	1.91982
$26$	0	0
$27$	2.13290	0.410477
$28$	0	0
$29$	2.17337	0.403585	0.201793	−	0.979428i	$-0.435323\pi$
$29$	2.17337	0.403585	0.201793	+	0.979428i	$0.435323\pi$
$30$	0	0
$31$	4.26026	0.765166	0.382583	−	0.923921i	$-0.375035\pi$
$31$	4.26026	0.765166	0.382583	+	0.923921i	$0.375035\pi$
$32$	0	0
$33$	−9.94257	−1.73078
$34$	0	0
$35$	5.78800	0.978350
$36$	0	0
$37$	7.41092	1.21835	0.609174	−	0.793037i	$-0.291501\pi$
$37$	7.41092	1.21835	0.609174	+	0.793037i	$0.291501\pi$
$38$	0	0
$39$	−12.1571	−1.94669
$40$	0	0
$41$	5.43554	0.848889	0.424445	−	0.905454i	$-0.360469\pi$
$41$	5.43554	0.848889	0.424445	+	0.905454i	$0.360469\pi$
$42$	0	0
$43$	7.88203	1.20200	0.600999	−	0.799250i	$-0.294770\pi$
$43$	7.88203	1.20200	0.600999	+	0.799250i	$0.294770\pi$
$44$	0	0
$45$	−7.83648	−1.16819
$46$	0	0
$47$	−8.32892	−1.21490	−0.607449	−	0.794359i	$-0.707807\pi$
$47$	−8.32892	−1.21490	−0.607449	+	0.794359i	$0.707807\pi$
$48$	0	0
$49$	−4.70528	−0.672183
$50$	0	0
$51$	−0.410247	−0.0574460
$52$	0	0
$53$	−4.23320	−0.581474	−0.290737	−	0.956803i	$-0.593901\pi$
$53$	−4.23320	−0.581474	−0.290737	+	0.956803i	$0.593901\pi$
$54$	0	0
$55$	−16.9034	−2.27926
$56$	0	0
$57$	−12.9169	−1.71089
$58$	0	0
$59$	−0.259033	−0.0337232	−0.0168616	−	0.999858i	$-0.505367\pi$
$59$	−0.259033	−0.0337232	−0.0168616	+	0.999858i	$0.505367\pi$
$60$	0	0
$61$	3.10398	0.397424	0.198712	−	0.980058i	$-0.436324\pi$
$61$	3.10398	0.397424	0.198712	+	0.980058i	$0.436324\pi$
$62$	0	0
$63$	−3.10687	−0.391428
$64$	0	0
$65$	−20.6684	−2.56360
$66$	0	0
$67$	9.04527	1.10506	0.552528	−	0.833494i	$-0.313663\pi$
$67$	9.04527	1.10506	0.552528	+	0.833494i	$0.313663\pi$
$68$	0	0
$69$	6.27258	0.755130
$70$	0	0
$71$	−5.83452	−0.692430	−0.346215	−	0.938155i	$-0.612533\pi$
$71$	−5.83452	−0.692430	−0.346215	+	0.938155i	$0.612533\pi$
$72$	0	0
$73$	8.42397	0.985951	0.492975	−	0.870043i	$-0.335909\pi$
$73$	8.42397	0.985951	0.492975	+	0.870043i	$0.335909\pi$
$74$	0	0
$75$	−21.5734	−2.49108
$76$	0	0
$77$	−6.70157	−0.763715
$78$	0	0
$79$	0.253285	0.0284968	0.0142484	−	0.999898i	$-0.495464\pi$
$79$	0.253285	0.0284968	0.0142484	+	0.999898i	$0.495464\pi$
$80$	0	0
$81$	−10.9464	−1.21627
$82$	0	0
$83$	3.29791	0.361993	0.180996	−	0.983484i	$-0.442068\pi$
$83$	3.29791	0.361993	0.180996	+	0.983484i	$0.442068\pi$
$84$	0	0
$85$	−0.697464	−0.0756505
$86$	0	0
$87$	−4.88452	−0.523675
$88$	0	0
$89$	10.5409	1.11734	0.558668	−	0.829391i	$-0.311313\pi$
$89$	10.5409	1.11734	0.558668	+	0.829391i	$0.311313\pi$
$90$	0	0
$91$	−8.19422	−0.858988
$92$	0	0
$93$	−9.57466	−0.992846
$94$	0	0
$95$	−21.9602	−2.25307
$96$	0	0
$97$	1.40835	0.142996	0.0714979	−	0.997441i	$-0.477222\pi$
$97$	1.40835	0.142996	0.0714979	+	0.997441i	$0.477222\pi$
$98$	0	0
$99$	9.07338	0.911909
$100$	0	0
$101$	−5.14574	−0.512021	−0.256010	−	0.966674i	$-0.582408\pi$
$101$	−5.14574	−0.512021	−0.256010	+	0.966674i	$0.582408\pi$
$102$	0	0
$103$	3.45878	0.340804	0.170402	−	0.985375i	$-0.445493\pi$
$103$	3.45878	0.340804	0.170402	+	0.985375i	$0.445493\pi$
$104$	0	0
$105$	−13.0081	−1.26947
$106$	0	0
$107$	−8.16307	−0.789153	−0.394577	−	0.918863i	$-0.629109\pi$
$107$	−8.16307	−0.789153	−0.394577	+	0.918863i	$0.629109\pi$
$108$	0	0
$109$	−10.5507	−1.01058	−0.505288	−	0.862950i	$-0.668614\pi$
$109$	−10.5507	−1.01058	−0.505288	+	0.862950i	$0.668614\pi$
$110$	0	0
$111$	−16.6555	−1.58087
$112$	0	0
$113$	5.54315	0.521455	0.260728	−	0.965412i	$-0.416038\pi$
$113$	5.54315	0.521455	0.260728	+	0.965412i	$0.416038\pi$
$114$	0	0
$115$	10.6641	0.994429
$116$	0	0
$117$	11.0943	1.02567
$118$	0	0
$119$	−0.276518	−0.0253484
$120$	0	0
$121$	8.57148	0.779225
$122$	0	0
$123$	−12.2160	−1.10148
$124$	0	0
$125$	−17.5727	−1.57175
$126$	0	0
$127$	2.16072	0.191733	0.0958666	−	0.995394i	$-0.469438\pi$
$127$	2.16072	0.191733	0.0958666	+	0.995394i	$0.469438\pi$
$128$	0	0
$129$	−17.7143	−1.55966
$130$	0	0
$131$	−6.39761	−0.558962	−0.279481	−	0.960151i	$-0.590162\pi$
$131$	−6.39761	−0.558962	−0.279481	+	0.960151i	$0.590162\pi$
$132$	0	0
$133$	−8.70637	−0.754938
$134$	0	0
$135$	−8.14956	−0.701403
$136$	0	0
$137$	15.6712	1.33888	0.669439	−	0.742867i	$-0.266535\pi$
$137$	15.6712	1.33888	0.669439	+	0.742867i	$0.266535\pi$
$138$	0	0
$139$	0.988483	0.0838420	0.0419210	−	0.999121i	$-0.486652\pi$
$139$	0.988483	0.0838420	0.0419210	+	0.999121i	$0.486652\pi$
$140$	0	0
$141$	18.7187	1.57640
$142$	0	0
$143$	23.9307	2.00118
$144$	0	0
$145$	−8.30420	−0.689626
$146$	0	0
$147$	10.5748	0.872195
$148$	0	0
$149$	−12.7506	−1.04457	−0.522286	−	0.852770i	$-0.674921\pi$
$149$	−12.7506	−1.04457	−0.522286	+	0.852770i	$0.674921\pi$
$150$	0	0
$151$	5.67305	0.461666	0.230833	−	0.972993i	$-0.425855\pi$
$151$	5.67305	0.461666	0.230833	+	0.972993i	$0.425855\pi$
$152$	0	0
$153$	0.374383	0.0302670
$154$	0	0
$155$	−16.2779	−1.30748
$156$	0	0
$157$	−12.4616	−0.994545	−0.497272	−	0.867594i	$-0.665665\pi$
$157$	−12.4616	−0.994545	−0.497272	+	0.867594i	$0.665665\pi$
$158$	0	0
$159$	9.51383	0.754496
$160$	0	0
$161$	4.22790	0.333205
$162$	0	0
$163$	16.1113	1.26193	0.630967	−	0.775810i	$-0.282658\pi$
$163$	16.1113	1.26193	0.630967	+	0.775810i	$0.282658\pi$
$164$	0	0
$165$	37.9894	2.95747
$166$	0	0
$167$	19.6750	1.52250	0.761248	−	0.648461i	$-0.224587\pi$
$167$	19.6750	1.52250	0.761248	+	0.648461i	$0.224587\pi$
$168$	0	0
$169$	16.2608	1.25083
$170$	0	0
$171$	11.7877	0.901429
$172$	0	0
$173$	−2.98403	−0.226872	−0.113436	−	0.993545i	$-0.536186\pi$
$173$	−2.98403	−0.226872	−0.113436	+	0.993545i	$0.536186\pi$
$174$	0	0
$175$	−14.5411	−1.09920
$176$	0	0
$177$	0.582159	0.0437578
$178$	0	0
$179$	−20.1599	−1.50682	−0.753412	−	0.657549i	$-0.771593\pi$
$179$	−20.1599	−1.50682	−0.753412	+	0.657549i	$0.771593\pi$
$180$	0	0
$181$	−15.0933	−1.12188	−0.560938	−	0.827858i	$-0.689560\pi$
$181$	−15.0933	−1.12188	−0.560938	+	0.827858i	$0.689560\pi$
$182$	0	0
$183$	−6.97599	−0.515680
$184$	0	0
$185$	−28.3162	−2.08185
$186$	0	0
$187$	0.807551	0.0590540
$188$	0	0
$189$	−3.23099	−0.235020
$190$	0	0
$191$	−10.3461	−0.748618	−0.374309	−	0.927304i	$-0.622120\pi$
$191$	−10.3461	−0.748618	−0.374309	+	0.927304i	$0.622120\pi$
$192$	0	0
$193$	1.59723	0.114971	0.0574856	−	0.998346i	$-0.481692\pi$
$193$	1.59723	0.114971	0.0574856	+	0.998346i	$0.481692\pi$
$194$	0	0
$195$	46.4508	3.32641
$196$	0	0
$197$	17.9177	1.27659	0.638293	−	0.769794i	$-0.279641\pi$
$197$	17.9177	1.27659	0.638293	+	0.769794i	$0.279641\pi$
$198$	0	0
$199$	−5.91697	−0.419443	−0.209722	−	0.977761i	$-0.567256\pi$
$199$	−5.91697	−0.419443	−0.209722	+	0.977761i	$0.567256\pi$
$200$	0	0
$201$	−20.3287	−1.43387
$202$	0	0
$203$	−3.29230	−0.231074
$204$	0	0
$205$	−20.7686	−1.45054
$206$	0	0
$207$	−5.72423	−0.397861
$208$	0	0
$209$	25.4264	1.75878
$210$	0	0
$211$	−12.5748	−0.865687	−0.432843	−	0.901469i	$-0.642490\pi$
$211$	−12.5748	−0.865687	−0.432843	+	0.901469i	$0.642490\pi$
$212$	0	0
$213$	13.1127	0.898468
$214$	0	0
$215$	−30.1163	−2.05391
$216$	0	0
$217$	−6.45359	−0.438098
$218$	0	0
$219$	−18.9323	−1.27933
$220$	0	0
$221$	0.987418	0.0664209
$222$	0	0
$223$	11.6207	0.778179	0.389089	−	0.921200i	$-0.372790\pi$
$223$	11.6207	0.778179	0.389089	+	0.921200i	$0.372790\pi$
$224$	0	0
$225$	19.6874	1.31250
$226$	0	0
$227$	13.5730	0.900872	0.450436	−	0.892809i	$-0.351269\pi$
$227$	13.5730	0.900872	0.450436	+	0.892809i	$0.351269\pi$
$228$	0	0
$229$	6.75440	0.446343	0.223172	−	0.974779i	$-0.428359\pi$
$229$	6.75440	0.446343	0.223172	+	0.974779i	$0.428359\pi$
$230$	0	0
$231$	15.0613	0.990964
$232$	0	0
$233$	8.14695	0.533725	0.266862	−	0.963735i	$-0.414013\pi$
$233$	8.14695	0.533725	0.266862	+	0.963735i	$0.414013\pi$
$234$	0	0
$235$	31.8238	2.07596
$236$	0	0
$237$	−0.569241	−0.0369762
$238$	0	0
$239$	−19.0179	−1.23016	−0.615082	−	0.788464i	$-0.710877\pi$
$239$	−19.0179	−1.23016	−0.615082	+	0.788464i	$0.710877\pi$
$240$	0	0
$241$	3.41645	0.220073	0.110036	−	0.993928i	$-0.464903\pi$
$241$	3.41645	0.220073	0.110036	+	0.993928i	$0.464903\pi$
$242$	0	0
$243$	18.2027	1.16770
$244$	0	0
$245$	17.9783	1.14859
$246$	0	0
$247$	31.0896	1.97818
$248$	0	0
$249$	−7.41184	−0.469706
$250$	0	0
$251$	30.1157	1.90089	0.950444	−	0.310895i	$-0.100629\pi$
$251$	30.1157	1.90089	0.950444	+	0.310895i	$0.100629\pi$
$252$	0	0
$253$	−12.3473	−0.776267
$254$	0	0
$255$	1.56750	0.0981609
$256$	0	0
$257$	−23.9337	−1.49294	−0.746471	−	0.665417i	$-0.768254\pi$
$257$	−23.9337	−1.49294	−0.746471	+	0.665417i	$0.768254\pi$
$258$	0	0
$259$	−11.2263	−0.697569
$260$	0	0
$261$	4.45751	0.275913
$262$	0	0
$263$	28.1589	1.73635	0.868177	−	0.496254i	$-0.165291\pi$
$263$	28.1589	1.73635	0.868177	+	0.496254i	$0.165291\pi$
$264$	0	0
$265$	16.1745	0.993594
$266$	0	0
$267$	−23.6901	−1.44981
$268$	0	0
$269$	7.69099	0.468928	0.234464	−	0.972125i	$-0.424667\pi$
$269$	7.69099	0.468928	0.234464	+	0.972125i	$0.424667\pi$
$270$	0	0
$271$	−17.3109	−1.05157	−0.525783	−	0.850619i	$-0.676227\pi$
$271$	−17.3109	−1.05157	−0.525783	+	0.850619i	$0.676227\pi$
$272$	0	0
$273$	18.4160	1.11459
$274$	0	0
$275$	42.4662	2.56081
$276$	0	0
$277$	3.74994	0.225312	0.112656	−	0.993634i	$-0.464064\pi$
$277$	3.74994	0.225312	0.112656	+	0.993634i	$0.464064\pi$
$278$	0	0
$279$	8.73763	0.523108
$280$	0	0
$281$	−13.0056	−0.775850	−0.387925	−	0.921691i	$-0.626808\pi$
$281$	−13.0056	−0.775850	−0.387925	+	0.921691i	$0.626808\pi$
$282$	0	0
$283$	9.90647	0.588879	0.294439	−	0.955670i	$-0.404867\pi$
$283$	9.90647	0.588879	0.294439	+	0.955670i	$0.404867\pi$
$284$	0	0
$285$	49.3540	2.92348
$286$	0	0
$287$	−8.23394	−0.486034
$288$	0	0
$289$	−16.9667	−0.998040
$290$	0	0
$291$	−3.16516	−0.185545
$292$	0	0
$293$	11.7930	0.688956	0.344478	−	0.938794i	$-0.388056\pi$
$293$	11.7930	0.688956	0.344478	+	0.938794i	$0.388056\pi$
$294$	0	0
$295$	0.989733	0.0576245
$296$	0	0
$297$	9.43589	0.547526
$298$	0	0
$299$	−15.0974	−0.873105
$300$	0	0
$301$	−11.9400	−0.688208
$302$	0	0
$303$	11.5647	0.664376
$304$	0	0
$305$	−11.8599	−0.679098
$306$	0	0
$307$	17.8243	1.01729	0.508644	−	0.860977i	$-0.330147\pi$
$307$	17.8243	1.01729	0.508644	+	0.860977i	$0.330147\pi$
$308$	0	0
$309$	−7.77339	−0.442213
$310$	0	0
$311$	−1.39632	−0.0791779	−0.0395890	−	0.999216i	$-0.512605\pi$
$311$	−1.39632	−0.0791779	−0.0395890	+	0.999216i	$0.512605\pi$
$312$	0	0
$313$	−5.56872	−0.314762	−0.157381	−	0.987538i	$-0.550305\pi$
$313$	−5.56872	−0.314762	−0.157381	+	0.987538i	$0.550305\pi$
$314$	0	0
$315$	11.8710	0.668853
$316$	0	0
$317$	−12.5299	−0.703747	−0.351873	−	0.936048i	$-0.614455\pi$
$317$	−12.5299	−0.703747	−0.351873	+	0.936048i	$0.614455\pi$
$318$	0	0
$319$	9.61493	0.538333
$320$	0	0
$321$	18.3460	1.02397
$322$	0	0
$323$	1.04913	0.0583753
$324$	0	0
$325$	51.9247	2.88027
$326$	0	0
$327$	23.7121	1.31128
$328$	0	0
$329$	12.6169	0.695594
$330$	0	0
$331$	32.6783	1.79616	0.898082	−	0.439828i	$-0.144961\pi$
$331$	32.6783	1.79616	0.898082	+	0.439828i	$0.144961\pi$
$332$	0	0
$333$	15.1995	0.832928
$334$	0	0
$335$	−34.5609	−1.88826
$336$	0	0
$337$	24.1418	1.31509	0.657544	−	0.753416i	$-0.271595\pi$
$337$	24.1418	1.31509	0.657544	+	0.753416i	$0.271595\pi$
$338$	0	0
$339$	−12.4579	−0.676618
$340$	0	0
$341$	18.8473	1.02064
$342$	0	0
$343$	17.7316	0.957414
$344$	0	0
$345$	−23.9668	−1.29033
$346$	0	0
$347$	−5.78658	−0.310640	−0.155320	−	0.987864i	$-0.549641\pi$
$347$	−5.78658	−0.310640	−0.155320	+	0.987864i	$0.549641\pi$
$348$	0	0
$349$	4.90191	0.262393	0.131197	−	0.991356i	$-0.458118\pi$
$349$	4.90191	0.262393	0.131197	+	0.991356i	$0.458118\pi$
$350$	0	0
$351$	11.5376	0.615829
$352$	0	0
$353$	27.0517	1.43981	0.719907	−	0.694070i	$-0.244184\pi$
$353$	27.0517	1.43981	0.719907	+	0.694070i	$0.244184\pi$
$354$	0	0
$355$	22.2930	1.18319
$356$	0	0
$357$	0.621456	0.0328909
$358$	0	0
$359$	18.2001	0.960566	0.480283	−	0.877113i	$-0.340534\pi$
$359$	18.2001	0.960566	0.480283	+	0.877113i	$0.340534\pi$
$360$	0	0
$361$	14.0327	0.738565
$362$	0	0
$363$	−19.2638	−1.01109
$364$	0	0
$365$	−32.1870	−1.68474
$366$	0	0
$367$	5.36891	0.280255	0.140127	−	0.990133i	$-0.455249\pi$
$367$	5.36891	0.280255	0.140127	+	0.990133i	$0.455249\pi$
$368$	0	0
$369$	11.1481	0.580346
$370$	0	0
$371$	6.41259	0.332925
$372$	0	0
$373$	−8.38422	−0.434119	−0.217059	−	0.976158i	$-0.569647\pi$
$373$	−8.38422	−0.434119	−0.217059	+	0.976158i	$0.569647\pi$
$374$	0	0
$375$	39.4935	2.03943
$376$	0	0
$377$	11.7565	0.605490
$378$	0	0
$379$	−19.6000	−1.00678	−0.503391	−	0.864059i	$-0.667915\pi$
$379$	−19.6000	−1.00678	−0.503391	+	0.864059i	$0.667915\pi$
$380$	0	0
$381$	−4.85608	−0.248785
$382$	0	0
$383$	−5.31418	−0.271542	−0.135771	−	0.990740i	$-0.543351\pi$
$383$	−5.31418	−0.271542	−0.135771	+	0.990740i	$0.543351\pi$
$384$	0	0
$385$	25.6059	1.30500
$386$	0	0
$387$	16.1657	0.821751
$388$	0	0
$389$	−13.9877	−0.709204	−0.354602	−	0.935017i	$-0.615384\pi$
$389$	−13.9877	−0.709204	−0.354602	+	0.935017i	$0.615384\pi$
$390$	0	0
$391$	−0.509469	−0.0257649
$392$	0	0
$393$	14.3782	0.725285
$394$	0	0
$395$	−0.967771	−0.0486938
$396$	0	0
$397$	−21.9695	−1.10262	−0.551310	−	0.834301i	$-0.685872\pi$
$397$	−21.9695	−1.10262	−0.551310	+	0.834301i	$0.685872\pi$
$398$	0	0
$399$	19.5670	0.979575
$400$	0	0
$401$	25.2392	1.26039	0.630193	−	0.776438i	$-0.282976\pi$
$401$	25.2392	1.26039	0.630193	+	0.776438i	$0.282976\pi$
$402$	0	0
$403$	23.0451	1.14796
$404$	0	0
$405$	41.8250	2.07830
$406$	0	0
$407$	32.7856	1.62512
$408$	0	0
$409$	2.53646	0.125420	0.0627098	−	0.998032i	$-0.480026\pi$
$409$	2.53646	0.125420	0.0627098	+	0.998032i	$0.480026\pi$
$410$	0	0
$411$	−35.2199	−1.73727
$412$	0	0
$413$	0.392392	0.0193083
$414$	0	0
$415$	−12.6009	−0.618555
$416$	0	0
$417$	−2.22155	−0.108790
$418$	0	0
$419$	−28.9487	−1.41424	−0.707118	−	0.707096i	$-0.750005\pi$
$419$	−28.9487	−1.41424	−0.707118	+	0.707096i	$0.750005\pi$
$420$	0	0
$421$	−3.63856	−0.177333	−0.0886664	−	0.996061i	$-0.528260\pi$
$421$	−3.63856	−0.177333	−0.0886664	+	0.996061i	$0.528260\pi$
$422$	0	0
$423$	−17.0823	−0.830569
$424$	0	0
$425$	1.75222	0.0849953
$426$	0	0
$427$	−4.70202	−0.227547
$428$	0	0
$429$	−53.7826	−2.59665
$430$	0	0
$431$	6.36696	0.306686	0.153343	−	0.988173i	$-0.450996\pi$
$431$	6.36696	0.306686	0.153343	+	0.988173i	$0.450996\pi$
$432$	0	0
$433$	−28.9777	−1.39258	−0.696290	−	0.717761i	$-0.745167\pi$
$433$	−28.9777	−1.39258	−0.696290	+	0.717761i	$0.745167\pi$
$434$	0	0
$435$	18.6631	0.894829
$436$	0	0
$437$	−16.0410	−0.767346
$438$	0	0
$439$	23.8979	1.14058	0.570292	−	0.821442i	$-0.306830\pi$
$439$	23.8979	1.14058	0.570292	+	0.821442i	$0.306830\pi$
$440$	0	0
$441$	−9.65034	−0.459540
$442$	0	0
$443$	−38.7096	−1.83915	−0.919575	−	0.392915i	$-0.871467\pi$
$443$	−38.7096	−1.83915	−0.919575	+	0.392915i	$0.871467\pi$
$444$	0	0
$445$	−40.2756	−1.90925
$446$	0	0
$447$	28.6562	1.35539
$448$	0	0
$449$	26.1299	1.23315	0.616574	−	0.787297i	$-0.288520\pi$
$449$	26.1299	1.23315	0.616574	+	0.787297i	$0.288520\pi$
$450$	0	0
$451$	24.0467	1.13231
$452$	0	0
$453$	−12.7498	−0.599038
$454$	0	0
$455$	31.3091	1.46780
$456$	0	0
$457$	−33.7819	−1.58025	−0.790125	−	0.612945i	$-0.789985\pi$
$457$	−33.7819	−1.58025	−0.790125	+	0.612945i	$0.789985\pi$
$458$	0	0
$459$	0.389340	0.0181728
$460$	0	0
$461$	−14.3864	−0.670041	−0.335020	−	0.942211i	$-0.608743\pi$
$461$	−14.3864	−0.670041	−0.335020	+	0.942211i	$0.608743\pi$
$462$	0	0
$463$	−12.3026	−0.571748	−0.285874	−	0.958267i	$-0.592284\pi$
$463$	−12.3026	−0.571748	−0.285874	+	0.958267i	$0.592284\pi$
$464$	0	0
$465$	36.5836	1.69652
$466$	0	0
$467$	28.3136	1.31020	0.655099	−	0.755543i	$-0.272627\pi$
$467$	28.3136	1.31020	0.655099	+	0.755543i	$0.272627\pi$
$468$	0	0
$469$	−13.7021	−0.632704
$470$	0	0
$471$	28.0067	1.29048
$472$	0	0
$473$	34.8698	1.60332
$474$	0	0
$475$	55.1701	2.53138
$476$	0	0
$477$	−8.68212	−0.397527
$478$	0	0
$479$	34.8281	1.59134	0.795668	−	0.605733i	$-0.207120\pi$
$479$	34.8281	1.59134	0.795668	+	0.605733i	$0.207120\pi$
$480$	0	0
$481$	40.0880	1.82786
$482$	0	0
$483$	−9.50192	−0.432353
$484$	0	0
$485$	−5.38112	−0.244344
$486$	0	0
$487$	20.3644	0.922797	0.461399	−	0.887193i	$-0.347348\pi$
$487$	20.3644	0.922797	0.461399	+	0.887193i	$0.347348\pi$
$488$	0	0
$489$	−36.2091	−1.63743
$490$	0	0
$491$	−7.94761	−0.358671	−0.179335	−	0.983788i	$-0.557395\pi$
$491$	−7.94761	−0.358671	−0.179335	+	0.983788i	$0.557395\pi$
$492$	0	0
$493$	0.396728	0.0178677
$494$	0	0
$495$	−34.6683	−1.55822
$496$	0	0
$497$	8.83833	0.396453
$498$	0	0
$499$	−32.2072	−1.44179	−0.720895	−	0.693044i	$-0.756269\pi$
$499$	−32.2072	−1.44179	−0.720895	+	0.693044i	$0.756269\pi$
$500$	0	0
$501$	−44.2182	−1.97552
$502$	0	0
$503$	6.26439	0.279315	0.139658	−	0.990200i	$-0.455400\pi$
$503$	6.26439	0.279315	0.139658	+	0.990200i	$0.455400\pi$
$504$	0	0
$505$	19.6613	0.874915
$506$	0	0
$507$	−36.5450	−1.62302
$508$	0	0
$509$	−5.78256	−0.256308	−0.128154	−	0.991754i	$-0.540905\pi$
$509$	−5.78256	−0.256308	−0.128154	+	0.991754i	$0.540905\pi$
$510$	0	0
$511$	−12.7609	−0.564510
$512$	0	0
$513$	12.2587	0.541233
$514$	0	0
$515$	−13.2156	−0.582349
$516$	0	0
$517$	−36.8469	−1.62052
$518$	0	0
$519$	6.70642	0.294379
$520$	0	0
$521$	−27.4610	−1.20309	−0.601545	−	0.798839i	$-0.705448\pi$
$521$	−27.4610	−1.20309	−0.601545	+	0.798839i	$0.705448\pi$
$522$	0	0
$523$	−17.3560	−0.758927	−0.379463	−	0.925207i	$-0.623891\pi$
$523$	−17.3560	−0.758927	−0.379463	+	0.925207i	$0.623891\pi$
$524$	0	0
$525$	32.6801	1.42628
$526$	0	0
$527$	0.777669	0.0338758
$528$	0	0
$529$	−15.2103	−0.661319
$530$	0	0
$531$	−0.531266	−0.0230550
$532$	0	0
$533$	29.4026	1.27357
$534$	0	0
$535$	31.1901	1.34847
$536$	0	0
$537$	45.3081	1.95519
$538$	0	0
$539$	−20.8160	−0.896608
$540$	0	0
$541$	12.8149	0.550955	0.275478	−	0.961307i	$-0.411164\pi$
$541$	12.8149	0.550955	0.275478	+	0.961307i	$0.411164\pi$
$542$	0	0
$543$	33.9212	1.45570
$544$	0	0
$545$	40.3131	1.72682
$546$	0	0
$547$	1.32391	0.0566065	0.0283032	−	0.999599i	$-0.490990\pi$
$547$	1.32391	0.0566065	0.0283032	+	0.999599i	$0.490990\pi$
$548$	0	0
$549$	6.36615	0.271700
$550$	0	0
$551$	12.4913	0.532146
$552$	0	0
$553$	−0.383684	−0.0163159
$554$	0	0
$555$	63.6388	2.70132
$556$	0	0
$557$	−21.5571	−0.913404	−0.456702	−	0.889620i	$-0.650969\pi$
$557$	−21.5571	−0.913404	−0.456702	+	0.889620i	$0.650969\pi$
$558$	0	0
$559$	42.6364	1.80333
$560$	0	0
$561$	−1.81492	−0.0766259
$562$	0	0
$563$	−9.21860	−0.388518	−0.194259	−	0.980950i	$-0.562230\pi$
$563$	−9.21860	−0.388518	−0.194259	+	0.980950i	$0.562230\pi$
$564$	0	0
$565$	−21.1797	−0.891037
$566$	0	0
$567$	16.5820	0.696380
$568$	0	0
$569$	−32.0699	−1.34444	−0.672221	−	0.740351i	$-0.734659\pi$
$569$	−32.0699	−1.34444	−0.672221	+	0.740351i	$0.734659\pi$
$570$	0	0
$571$	24.0105	1.00481	0.502405	−	0.864632i	$-0.332449\pi$
$571$	24.0105	1.00481	0.502405	+	0.864632i	$0.332449\pi$
$572$	0	0
$573$	23.2522	0.971374
$574$	0	0
$575$	−26.7911	−1.11727
$576$	0	0
$577$	−31.5387	−1.31298	−0.656488	−	0.754337i	$-0.727959\pi$
$577$	−31.5387	−1.31298	−0.656488	+	0.754337i	$0.727959\pi$
$578$	0	0
$579$	−3.58967	−0.149182
$580$	0	0
$581$	−4.99579	−0.207260
$582$	0	0
$583$	−18.7275	−0.775615
$584$	0	0
$585$	−42.3900	−1.75261
$586$	0	0
$587$	0.297232	0.0122681	0.00613404	−	0.999981i	$-0.498047\pi$
$587$	0.297232	0.0122681	0.00613404	+	0.999981i	$0.498047\pi$
$588$	0	0
$589$	24.4855	1.00891
$590$	0	0
$591$	−40.2689	−1.65644
$592$	0	0
$593$	−0.754062	−0.0309656	−0.0154828	−	0.999880i	$-0.504929\pi$
$593$	−0.754062	−0.0309656	−0.0154828	+	0.999880i	$0.504929\pi$
$594$	0	0
$595$	1.05654	0.0433140
$596$	0	0
$597$	13.2980	0.544251
$598$	0	0
$599$	−23.7984	−0.972375	−0.486187	−	0.873855i	$-0.661613\pi$
$599$	−23.7984	−0.972375	−0.486187	+	0.873855i	$0.661613\pi$
$600$	0	0
$601$	−10.6085	−0.432730	−0.216365	−	0.976313i	$-0.569420\pi$
$601$	−10.6085	−0.432730	−0.216365	+	0.976313i	$0.569420\pi$
$602$	0	0
$603$	18.5515	0.755476
$604$	0	0
$605$	−32.7506	−1.33150
$606$	0	0
$607$	33.3665	1.35430	0.677152	−	0.735843i	$-0.263214\pi$
$607$	33.3665	1.35430	0.677152	+	0.735843i	$0.263214\pi$
$608$	0	0
$609$	7.39923	0.299832
$610$	0	0
$611$	−45.0538	−1.82268
$612$	0	0
$613$	−29.7669	−1.20228	−0.601138	−	0.799146i	$-0.705286\pi$
$613$	−29.7669	−1.20228	−0.601138	+	0.799146i	$0.705286\pi$
$614$	0	0
$615$	46.6760	1.88216
$616$	0	0
$617$	−42.6275	−1.71612	−0.858059	−	0.513551i	$-0.828330\pi$
$617$	−42.6275	−1.71612	−0.858059	+	0.513551i	$0.828330\pi$
$618$	0	0
$619$	−14.7432	−0.592581	−0.296291	−	0.955098i	$-0.595750\pi$
$619$	−14.7432	−0.592581	−0.296291	+	0.955098i	$0.595750\pi$
$620$	0	0
$621$	−5.95292	−0.238883
$622$	0	0
$623$	−15.9678	−0.639735
$624$	0	0
$625$	19.1475	0.765900
$626$	0	0
$627$	−57.1441	−2.28211
$628$	0	0
$629$	1.35279	0.0539392
$630$	0	0
$631$	2.96880	0.118186	0.0590931	−	0.998252i	$-0.481179\pi$
$631$	2.96880	0.118186	0.0590931	+	0.998252i	$0.481179\pi$
$632$	0	0
$633$	28.2611	1.12328
$634$	0	0
$635$	−8.25586	−0.327624
$636$	0	0
$637$	−25.4524	−1.00846
$638$	0	0
$639$	−11.9664	−0.473383
$640$	0	0
$641$	4.04405	0.159730	0.0798651	−	0.996806i	$-0.474551\pi$
$641$	4.04405	0.159730	0.0798651	+	0.996806i	$0.474551\pi$
$642$	0	0
$643$	39.4495	1.55574	0.777869	−	0.628427i	$-0.216301\pi$
$643$	39.4495	1.55574	0.777869	+	0.628427i	$0.216301\pi$
$644$	0	0
$645$	67.6844	2.66507
$646$	0	0
$647$	−10.3032	−0.405060	−0.202530	−	0.979276i	$-0.564916\pi$
$647$	−10.3032	−0.405060	−0.202530	+	0.979276i	$0.564916\pi$
$648$	0	0
$649$	−1.14595	−0.0449826
$650$	0	0
$651$	14.5040	0.568457
$652$	0	0
$653$	17.2549	0.675235	0.337618	−	0.941283i	$-0.390379\pi$
$653$	17.2549	0.675235	0.337618	+	0.941283i	$0.390379\pi$
$654$	0	0
$655$	24.4445	0.955126
$656$	0	0
$657$	17.2772	0.674049
$658$	0	0
$659$	27.5321	1.07250	0.536249	−	0.844060i	$-0.319841\pi$
$659$	27.5321	1.07250	0.536249	+	0.844060i	$0.319841\pi$
$660$	0	0
$661$	24.2337	0.942580	0.471290	−	0.881978i	$-0.343789\pi$
$661$	24.2337	0.942580	0.471290	+	0.881978i	$0.343789\pi$
$662$	0	0
$663$	−2.21916	−0.0861849
$664$	0	0
$665$	33.2660	1.29000
$666$	0	0
$667$	−6.06588	−0.234872
$668$	0	0
$669$	−26.1167	−1.00973
$670$	0	0
$671$	13.7319	0.530114
$672$	0	0
$673$	−4.98779	−0.192265	−0.0961325	−	0.995369i	$-0.530647\pi$
$673$	−4.98779	−0.192265	−0.0961325	+	0.995369i	$0.530647\pi$
$674$	0	0
$675$	20.4740	0.788044
$676$	0	0
$677$	−31.5720	−1.21341	−0.606705	−	0.794927i	$-0.707509\pi$
$677$	−31.5720	−1.21341	−0.606705	+	0.794927i	$0.707509\pi$
$678$	0	0
$679$	−2.13341	−0.0818728
$680$	0	0
$681$	−30.5044	−1.16893
$682$	0	0
$683$	33.6808	1.28876	0.644379	−	0.764706i	$-0.277116\pi$
$683$	33.6808	1.28876	0.644379	+	0.764706i	$0.277116\pi$
$684$	0	0
$685$	−59.8776	−2.28781
$686$	0	0
$687$	−15.1801	−0.579155
$688$	0	0
$689$	−22.8987	−0.872372
$690$	0	0
$691$	−27.8194	−1.05830	−0.529151	−	0.848528i	$-0.677489\pi$
$691$	−27.8194	−1.05830	−0.529151	+	0.848528i	$0.677489\pi$
$692$	0	0
$693$	−13.7447	−0.522117
$694$	0	0
$695$	−3.77687	−0.143265
$696$	0	0
$697$	0.992204	0.0375824
$698$	0	0
$699$	−18.3097	−0.692538
$700$	0	0
$701$	−16.3027	−0.615745	−0.307872	−	0.951428i	$-0.599617\pi$
$701$	−16.3027	−0.615745	−0.307872	+	0.951428i	$0.599617\pi$
$702$	0	0
$703$	42.5936	1.60645
$704$	0	0
$705$	−71.5219	−2.69367
$706$	0	0
$707$	7.79495	0.293159
$708$	0	0
$709$	6.28475	0.236029	0.118014	−	0.993012i	$-0.462347\pi$
$709$	6.28475	0.236029	0.118014	+	0.993012i	$0.462347\pi$
$710$	0	0
$711$	0.519477	0.0194819
$712$	0	0
$713$	−11.8904	−0.445298
$714$	0	0
$715$	−91.4362	−3.41952
$716$	0	0
$717$	42.7414	1.59621
$718$	0	0
$719$	27.6145	1.02985	0.514924	−	0.857236i	$-0.327820\pi$
$719$	27.6145	1.02985	0.514924	+	0.857236i	$0.327820\pi$
$720$	0	0
$721$	−5.23949	−0.195129
$722$	0	0
$723$	−7.67824	−0.285557
$724$	0	0
$725$	20.8625	0.774813
$726$	0	0
$727$	−39.9851	−1.48297	−0.741483	−	0.670972i	$-0.765877\pi$
$727$	−39.9851	−1.48297	−0.741483	+	0.670972i	$0.765877\pi$
$728$	0	0
$729$	−8.07006	−0.298891
$730$	0	0
$731$	1.43879	0.0532154
$732$	0	0
$733$	18.6625	0.689316	0.344658	−	0.938728i	$-0.387995\pi$
$733$	18.6625	0.689316	0.344658	+	0.938728i	$0.387995\pi$
$734$	0	0
$735$	−40.4050	−1.49036
$736$	0	0
$737$	40.0160	1.47401
$738$	0	0
$739$	12.1223	0.445927	0.222964	−	0.974827i	$-0.428427\pi$
$739$	12.1223	0.445927	0.222964	+	0.974827i	$0.428427\pi$
$740$	0	0
$741$	−69.8718	−2.56681
$742$	0	0
$743$	−11.0663	−0.405982	−0.202991	−	0.979181i	$-0.565066\pi$
$743$	−11.0663	−0.405982	−0.202991	+	0.979181i	$0.565066\pi$
$744$	0	0
$745$	48.7186	1.78491
$746$	0	0
$747$	6.76389	0.247478
$748$	0	0
$749$	12.3657	0.451833
$750$	0	0
$751$	9.41284	0.343479	0.171740	−	0.985142i	$-0.445061\pi$
$751$	9.41284	0.343479	0.171740	+	0.985142i	$0.445061\pi$
$752$	0	0
$753$	−67.6831	−2.46651
$754$	0	0
$755$	−21.6760	−0.788872
$756$	0	0
$757$	−10.4599	−0.380171	−0.190086	−	0.981768i	$-0.560877\pi$
$757$	−10.4599	−0.380171	−0.190086	+	0.981768i	$0.560877\pi$
$758$	0	0
$759$	27.7497	1.00725
$760$	0	0
$761$	31.6708	1.14807	0.574033	−	0.818832i	$-0.305378\pi$
$761$	31.6708	1.14807	0.574033	+	0.818832i	$0.305378\pi$
$762$	0	0
$763$	15.9826	0.578609
$764$	0	0
$765$	−1.43047	−0.0517188
$766$	0	0
$767$	−1.40119	−0.0505941
$768$	0	0
$769$	0.626394	0.0225883	0.0112942	−	0.999936i	$-0.496405\pi$
$769$	0.626394	0.0225883	0.0112942	+	0.999936i	$0.496405\pi$
$770$	0	0
$771$	53.7894	1.93718
$772$	0	0
$773$	13.0082	0.467871	0.233936	−	0.972252i	$-0.424839\pi$
$773$	13.0082	0.467871	0.233936	+	0.972252i	$0.424839\pi$
$774$	0	0
$775$	40.8948	1.46898
$776$	0	0
$777$	25.2304	0.905135
$778$	0	0
$779$	31.2403	1.11930
$780$	0	0
$781$	−25.8117	−0.923616
$782$	0	0
$783$	4.63559	0.165663
$784$	0	0
$785$	47.6143	1.69943
$786$	0	0
$787$	−28.5381	−1.01727	−0.508637	−	0.860981i	$-0.669851\pi$
$787$	−28.5381	−1.01727	−0.508637	+	0.860981i	$0.669851\pi$
$788$	0	0
$789$	−63.2854	−2.25302
$790$	0	0
$791$	−8.39695	−0.298561
$792$	0	0
$793$	16.7904	0.596246
$794$	0	0
$795$	−36.3512	−1.28924
$796$	0	0
$797$	−53.0810	−1.88022	−0.940112	−	0.340864i	$-0.889280\pi$
$797$	−53.0810	−1.88022	−0.940112	+	0.340864i	$0.889280\pi$
$798$	0	0
$799$	−1.52036	−0.0537865
$800$	0	0
$801$	21.6190	0.763871
$802$	0	0
$803$	37.2674	1.31514
$804$	0	0
$805$	−16.1543	−0.569364
$806$	0	0
$807$	−17.2850	−0.608461
$808$	0	0
$809$	−2.11514	−0.0743643	−0.0371821	−	0.999309i	$-0.511838\pi$
$809$	−2.11514	−0.0743643	−0.0371821	+	0.999309i	$0.511838\pi$
$810$	0	0
$811$	10.6009	0.372249	0.186124	−	0.982526i	$-0.440407\pi$
$811$	10.6009	0.372249	0.186124	+	0.982526i	$0.440407\pi$
$812$	0	0
$813$	38.9052	1.36447
$814$	0	0
$815$	−61.5593	−2.15633
$816$	0	0
$817$	45.3013	1.58489
$818$	0	0
$819$	−16.8060	−0.587250
$820$	0	0
$821$	5.66275	0.197631	0.0988156	−	0.995106i	$-0.468495\pi$
$821$	5.66275	0.197631	0.0988156	+	0.995106i	$0.468495\pi$
$822$	0	0
$823$	−11.9963	−0.418164	−0.209082	−	0.977898i	$-0.567048\pi$
$823$	−11.9963	−0.418164	−0.209082	+	0.977898i	$0.567048\pi$
$824$	0	0
$825$	−95.4400	−3.32279
$826$	0	0
$827$	25.8689	0.899549	0.449775	−	0.893142i	$-0.351504\pi$
$827$	25.8689	0.899549	0.449775	+	0.893142i	$0.351504\pi$
$828$	0	0
$829$	−3.76842	−0.130883	−0.0654413	−	0.997856i	$-0.520846\pi$
$829$	−3.76842	−0.130883	−0.0654413	+	0.997856i	$0.520846\pi$
$830$	0	0
$831$	−8.42775	−0.292355
$832$	0	0
$833$	−0.858902	−0.0297592
$834$	0	0
$835$	−75.1757	−2.60156
$836$	0	0
$837$	9.08672	0.314083
$838$	0	0
$839$	20.7153	0.715171	0.357586	−	0.933880i	$-0.383600\pi$
$839$	20.7153	0.715171	0.357586	+	0.933880i	$0.383600\pi$
$840$	0	0
$841$	−24.2764	−0.837119
$842$	0	0
$843$	29.2293	1.00671
$844$	0	0
$845$	−62.1304	−2.13735
$846$	0	0
$847$	−12.9844	−0.446148
$848$	0	0
$849$	−22.2641	−0.764104
$850$	0	0
$851$	−20.6839	−0.709033
$852$	0	0
$853$	4.50005	0.154079	0.0770395	−	0.997028i	$-0.475453\pi$
$853$	4.50005	0.154079	0.0770395	+	0.997028i	$0.475453\pi$
$854$	0	0
$855$	−45.0394	−1.54032
$856$	0	0
$857$	32.8896	1.12349	0.561744	−	0.827311i	$-0.310131\pi$
$857$	32.8896	1.12349	0.561744	+	0.827311i	$0.310131\pi$
$858$	0	0
$859$	34.2754	1.16946	0.584731	−	0.811227i	$-0.301200\pi$
$859$	34.2754	1.16946	0.584731	+	0.811227i	$0.301200\pi$
$860$	0	0
$861$	18.5052	0.630657
$862$	0	0
$863$	38.0262	1.29443	0.647213	−	0.762309i	$-0.275934\pi$
$863$	38.0262	1.29443	0.647213	+	0.762309i	$0.275934\pi$
$864$	0	0
$865$	11.4016	0.387667
$866$	0	0
$867$	38.1315	1.29501
$868$	0	0
$869$	1.12052	0.0380112
$870$	0	0
$871$	48.9288	1.65789
$872$	0	0
$873$	2.88846	0.0977596
$874$	0	0
$875$	26.6197	0.899910
$876$	0	0
$877$	−35.0370	−1.18311	−0.591557	−	0.806263i	$-0.701487\pi$
$877$	−35.0370	−1.18311	−0.591557	+	0.806263i	$0.701487\pi$
$878$	0	0
$879$	−26.5040	−0.893959
$880$	0	0
$881$	41.9284	1.41260	0.706302	−	0.707911i	$-0.250362\pi$
$881$	41.9284	1.41260	0.706302	+	0.707911i	$0.250362\pi$
$882$	0	0
$883$	16.7847	0.564851	0.282425	−	0.959289i	$-0.408861\pi$
$883$	16.7847	0.564851	0.282425	+	0.959289i	$0.408861\pi$
$884$	0	0
$885$	−2.22436	−0.0747711
$886$	0	0
$887$	57.6221	1.93476	0.967381	−	0.253326i	$-0.0815245\pi$
$887$	57.6221	1.93476	0.967381	+	0.253326i	$0.0815245\pi$
$888$	0	0
$889$	−3.27314	−0.109777
$890$	0	0
$891$	−48.4267	−1.62236
$892$	0	0
$893$	−47.8697	−1.60190
$894$	0	0
$895$	77.0286	2.57478
$896$	0	0
$897$	33.9304	1.13290
$898$	0	0
$899$	9.25914	0.308810
$900$	0	0
$901$	−0.772728	−0.0257433
$902$	0	0
$903$	26.8343	0.892989
$904$	0	0
$905$	57.6697	1.91701
$906$	0	0
$907$	−41.9502	−1.39293	−0.696467	−	0.717589i	$-0.745246\pi$
$907$	−41.9502	−1.39293	−0.696467	+	0.717589i	$0.745246\pi$
$908$	0	0
$909$	−10.5537	−0.350045
$910$	0	0
$911$	19.9560	0.661170	0.330585	−	0.943776i	$-0.392754\pi$
$911$	19.9560	0.661170	0.330585	+	0.943776i	$0.392754\pi$
$912$	0	0
$913$	14.5899	0.482854
$914$	0	0
$915$	26.6544	0.881168
$916$	0	0
$917$	9.69132	0.320036
$918$	0	0
$919$	−58.3467	−1.92468	−0.962339	−	0.271851i	$-0.912364\pi$
$919$	−58.3467	−1.92468	−0.962339	+	0.271851i	$0.912364\pi$
$920$	0	0
$921$	−40.0590	−1.31999
$922$	0	0
$923$	−31.5608	−1.03884
$924$	0	0
$925$	71.1383	2.33901
$926$	0	0
$927$	7.09383	0.232992
$928$	0	0
$929$	8.96024	0.293976	0.146988	−	0.989138i	$-0.453042\pi$
$929$	8.96024	0.293976	0.146988	+	0.989138i	$0.453042\pi$
$930$	0	0
$931$	−27.0432	−0.886304
$932$	0	0
$933$	3.13813	0.102738
$934$	0	0
$935$	−3.08556	−0.100908
$936$	0	0
$937$	15.6273	0.510522	0.255261	−	0.966872i	$-0.417839\pi$
$937$	15.6273	0.510522	0.255261	+	0.966872i	$0.417839\pi$
$938$	0	0
$939$	12.5153	0.408422
$940$	0	0
$941$	−0.524539	−0.0170995	−0.00854974	−	0.999963i	$-0.502721\pi$
$941$	−0.524539	−0.0170995	−0.00854974	+	0.999963i	$0.502721\pi$
$942$	0	0
$943$	−15.1706	−0.494022
$944$	0	0
$945$	12.3452	0.401590
$946$	0	0
$947$	13.4216	0.436143	0.218072	−	0.975933i	$-0.430023\pi$
$947$	13.4216	0.436143	0.218072	+	0.975933i	$0.430023\pi$
$948$	0	0
$949$	45.5680	1.47920
$950$	0	0
$951$	28.1600	0.913151
$952$	0	0
$953$	−15.4770	−0.501351	−0.250675	−	0.968071i	$-0.580653\pi$
$953$	−15.4770	−0.501351	−0.250675	+	0.968071i	$0.580653\pi$
$954$	0	0
$955$	39.5312	1.27920
$956$	0	0
$957$	−21.6089	−0.698518
$958$	0	0
$959$	−23.7392	−0.766579
$960$	0	0
$961$	−12.8502	−0.414521
$962$	0	0
$963$	−16.7421	−0.539508
$964$	0	0
$965$	−6.10283	−0.196457
$966$	0	0
$967$	60.0879	1.93230	0.966149	−	0.257985i	$-0.0830584\pi$
$967$	60.0879	1.93230	0.966149	+	0.257985i	$0.0830584\pi$
$968$	0	0
$969$	−2.35786	−0.0757453
$970$	0	0
$971$	−9.53742	−0.306070	−0.153035	−	0.988221i	$-0.548905\pi$
$971$	−9.53742	−0.306070	−0.153035	+	0.988221i	$0.548905\pi$
$972$	0	0
$973$	−1.49739	−0.0480041
$974$	0	0
$975$	−116.697	−3.73731
$976$	0	0
$977$	18.2162	0.582787	0.291394	−	0.956603i	$-0.405881\pi$
$977$	18.2162	0.582787	0.291394	+	0.956603i	$0.405881\pi$
$978$	0	0
$979$	46.6327	1.49039
$980$	0	0
$981$	−21.6391	−0.690885
$982$	0	0
$983$	−17.2556	−0.550370	−0.275185	−	0.961391i	$-0.588739\pi$
$983$	−17.2556	−0.550370	−0.275185	+	0.961391i	$0.588739\pi$
$984$	0	0
$985$	−68.4615	−2.18136
$986$	0	0
$987$	−28.3557	−0.902573
$988$	0	0
$989$	−21.9987	−0.699519
$990$	0	0
$991$	1.20829	0.0383827	0.0191913	−	0.999816i	$-0.493891\pi$
$991$	1.20829	0.0383827	0.0191913	+	0.999816i	$0.493891\pi$
$992$	0	0
$993$	−73.4424	−2.33062
$994$	0	0
$995$	22.6080	0.716723
$996$	0	0
$997$	46.3168	1.46687	0.733435	−	0.679760i	$-0.237916\pi$
$997$	46.3168	1.46687	0.733435	+	0.679760i	$0.237916\pi$
$998$	0	0
$999$	15.8068	0.500104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8012.2.a.b.1.14	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8012.2.a.b.1.14	✓	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)

\(f(q)\)	\(=\)	\(q-2.24743 q^{3} -3.82088 q^{5} -1.51483 q^{7} +2.05096 q^{9} +O(q^{10})\)	\(q-2.24743 q^{3} -3.82088 q^{5} -1.51483 q^{7} +2.05096 q^{9} +4.42397 q^{11} +5.40932 q^{13} +8.58718 q^{15} +0.182540 q^{17} +5.74741 q^{19} +3.40449 q^{21} -2.79100 q^{23} +9.59912 q^{25} +2.13290 q^{27} +2.17337 q^{29} +4.26026 q^{31} -9.94257 q^{33} +5.78800 q^{35} +7.41092 q^{37} -12.1571 q^{39} +5.43554 q^{41} +7.88203 q^{43} -7.83648 q^{45} -8.32892 q^{47} -4.70528 q^{49} -0.410247 q^{51} -4.23320 q^{53} -16.9034 q^{55} -12.9169 q^{57} -0.259033 q^{59} +3.10398 q^{61} -3.10687 q^{63} -20.6684 q^{65} +9.04527 q^{67} +6.27258 q^{69} -5.83452 q^{71} +8.42397 q^{73} -21.5734 q^{75} -6.70157 q^{77} +0.253285 q^{79} -10.9464 q^{81} +3.29791 q^{83} -0.697464 q^{85} -4.88452 q^{87} +10.5409 q^{89} -8.19422 q^{91} -9.57466 q^{93} -21.9602 q^{95} +1.40835 q^{97} +9.07338 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) \) 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