LMFDB - Embedded newform 8012.2.a.b.1.15

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			1.15
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)

$f(q)$	$=$	$q-2.08516 q^{3} +1.63979 q^{5} +1.21456 q^{7} +1.34791 q^{9} +O(q^{10})$	$q-2.08516 q^{3} +1.63979 q^{5} +1.21456 q^{7} +1.34791 q^{9} -5.36395 q^{11} -1.23888 q^{13} -3.41924 q^{15} +7.24629 q^{17} +4.70626 q^{19} -2.53256 q^{21} +4.91032 q^{23} -2.31108 q^{25} +3.44489 q^{27} -0.253647 q^{29} +3.63592 q^{31} +11.1847 q^{33} +1.99163 q^{35} -8.66068 q^{37} +2.58328 q^{39} +7.17763 q^{41} -6.78317 q^{43} +2.21029 q^{45} +4.34068 q^{47} -5.52484 q^{49} -15.1097 q^{51} -2.42254 q^{53} -8.79577 q^{55} -9.81331 q^{57} +3.96048 q^{59} -15.1004 q^{61} +1.63712 q^{63} -2.03152 q^{65} +7.40337 q^{67} -10.2388 q^{69} +10.9067 q^{71} -6.50048 q^{73} +4.81897 q^{75} -6.51486 q^{77} +5.60274 q^{79} -11.2269 q^{81} -4.83564 q^{83} +11.8824 q^{85} +0.528894 q^{87} +15.1448 q^{89} -1.50470 q^{91} -7.58148 q^{93} +7.71729 q^{95} +3.51891 q^{97} -7.23010 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.08516	−1.20387	−0.601935	−	0.798545i	$-0.705603\pi$
$3$	−2.08516	−1.20387	−0.601935	+	0.798545i	$0.705603\pi$
$4$	0	0
$5$	1.63979	0.733338	0.366669	−	0.930351i	$-0.380498\pi$
$5$	1.63979	0.733338	0.366669	+	0.930351i	$0.380498\pi$
$6$	0	0
$7$	1.21456	0.459062	0.229531	−	0.973301i	$-0.426281\pi$
$7$	1.21456	0.459062	0.229531	+	0.973301i	$0.426281\pi$
$8$	0	0
$9$	1.34791	0.449302
$10$	0	0
$11$	−5.36395	−1.61729	−0.808646	−	0.588296i	$-0.799799\pi$
$11$	−5.36395	−1.61729	−0.808646	+	0.588296i	$0.799799\pi$
$12$	0	0
$13$	−1.23888	−0.343605	−0.171802	−	0.985131i	$-0.554959\pi$
$13$	−1.23888	−0.343605	−0.171802	+	0.985131i	$0.554959\pi$
$14$	0	0
$15$	−3.41924	−0.882843
$16$	0	0
$17$	7.24629	1.75748	0.878742	−	0.477297i	$-0.158383\pi$
$17$	7.24629	1.75748	0.878742	+	0.477297i	$0.158383\pi$
$18$	0	0
$19$	4.70626	1.07969	0.539845	−	0.841765i	$-0.318483\pi$
$19$	4.70626	1.07969	0.539845	+	0.841765i	$0.318483\pi$
$20$	0	0
$21$	−2.53256	−0.552651
$22$	0	0
$23$	4.91032	1.02387	0.511937	−	0.859023i	$-0.328928\pi$
$23$	4.91032	1.02387	0.511937	+	0.859023i	$0.328928\pi$
$24$	0	0
$25$	−2.31108	−0.462215
$26$	0	0
$27$	3.44489	0.662968
$28$	0	0
$29$	−0.253647	−0.0471010	−0.0235505	−	0.999723i	$-0.507497\pi$
$29$	−0.253647	−0.0471010	−0.0235505	+	0.999723i	$0.507497\pi$
$30$	0	0
$31$	3.63592	0.653030	0.326515	−	0.945192i	$-0.394126\pi$
$31$	3.63592	0.653030	0.326515	+	0.945192i	$0.394126\pi$
$32$	0	0
$33$	11.1847	1.94701
$34$	0	0
$35$	1.99163	0.336647
$36$	0	0
$37$	−8.66068	−1.42381	−0.711904	−	0.702277i	$-0.752167\pi$
$37$	−8.66068	−1.42381	−0.711904	+	0.702277i	$0.752167\pi$
$38$	0	0
$39$	2.58328	0.413655
$40$	0	0
$41$	7.17763	1.12096	0.560479	−	0.828169i	$-0.310617\pi$
$41$	7.17763	1.12096	0.560479	+	0.828169i	$0.310617\pi$
$42$	0	0
$43$	−6.78317	−1.03442	−0.517212	−	0.855857i	$-0.673030\pi$
$43$	−6.78317	−1.03442	−0.517212	+	0.855857i	$0.673030\pi$
$44$	0	0
$45$	2.21029	0.329490
$46$	0	0
$47$	4.34068	0.633153	0.316577	−	0.948567i	$-0.397467\pi$
$47$	4.34068	0.633153	0.316577	+	0.948567i	$0.397467\pi$
$48$	0	0
$49$	−5.52484	−0.789262
$50$	0	0
$51$	−15.1097	−2.11578
$52$	0	0
$53$	−2.42254	−0.332762	−0.166381	−	0.986062i	$-0.553208\pi$
$53$	−2.42254	−0.332762	−0.166381	+	0.986062i	$0.553208\pi$
$54$	0	0
$55$	−8.79577	−1.18602
$56$	0	0
$57$	−9.81331	−1.29981
$58$	0	0
$59$	3.96048	0.515610	0.257805	−	0.966197i	$-0.417001\pi$
$59$	3.96048	0.515610	0.257805	+	0.966197i	$0.417001\pi$
$60$	0	0
$61$	−15.1004	−1.93341	−0.966707	−	0.255887i	$-0.917633\pi$
$61$	−15.1004	−1.93341	−0.966707	+	0.255887i	$0.917633\pi$
$62$	0	0
$63$	1.63712	0.206257
$64$	0	0
$65$	−2.03152	−0.251978
$66$	0	0
$67$	7.40337	0.904466	0.452233	−	0.891900i	$-0.350628\pi$
$67$	7.40337	0.904466	0.452233	+	0.891900i	$0.350628\pi$
$68$	0	0
$69$	−10.2388	−1.23261
$70$	0	0
$71$	10.9067	1.29439	0.647196	−	0.762324i	$-0.275942\pi$
$71$	10.9067	1.29439	0.647196	+	0.762324i	$0.275942\pi$
$72$	0	0
$73$	−6.50048	−0.760823	−0.380412	−	0.924817i	$-0.624218\pi$
$73$	−6.50048	−0.760823	−0.380412	+	0.924817i	$0.624218\pi$
$74$	0	0
$75$	4.81897	0.556447
$76$	0	0
$77$	−6.51486	−0.742437
$78$	0	0
$79$	5.60274	0.630357	0.315179	−	0.949032i	$-0.397936\pi$
$79$	5.60274	0.630357	0.315179	+	0.949032i	$0.397936\pi$
$80$	0	0
$81$	−11.2269	−1.24743
$82$	0	0
$83$	−4.83564	−0.530780	−0.265390	−	0.964141i	$-0.585501\pi$
$83$	−4.83564	−0.530780	−0.265390	+	0.964141i	$0.585501\pi$
$84$	0	0
$85$	11.8824	1.28883
$86$	0	0
$87$	0.528894	0.0567034
$88$	0	0
$89$	15.1448	1.60535	0.802674	−	0.596417i	$-0.203410\pi$
$89$	15.1448	1.60535	0.802674	+	0.596417i	$0.203410\pi$
$90$	0	0
$91$	−1.50470	−0.157736
$92$	0	0
$93$	−7.58148	−0.786163
$94$	0	0
$95$	7.71729	0.791777
$96$	0	0
$97$	3.51891	0.357292	0.178646	−	0.983913i	$-0.442828\pi$
$97$	3.51891	0.357292	0.178646	+	0.983913i	$0.442828\pi$
$98$	0	0
$99$	−7.23010	−0.726653
$100$	0	0
$101$	−11.3306	−1.12743	−0.563716	−	0.825968i	$-0.690629\pi$
$101$	−11.3306	−1.12743	−0.563716	+	0.825968i	$0.690629\pi$
$102$	0	0
$103$	−0.294945	−0.0290618	−0.0145309	−	0.999894i	$-0.504625\pi$
$103$	−0.294945	−0.0290618	−0.0145309	+	0.999894i	$0.504625\pi$
$104$	0	0
$105$	−4.15288	−0.405280
$106$	0	0
$107$	13.1222	1.26857	0.634286	−	0.773099i	$-0.281294\pi$
$107$	13.1222	1.26857	0.634286	+	0.773099i	$0.281294\pi$
$108$	0	0
$109$	−4.53173	−0.434061	−0.217030	−	0.976165i	$-0.569637\pi$
$109$	−4.53173	−0.434061	−0.217030	+	0.976165i	$0.569637\pi$
$110$	0	0
$111$	18.0589	1.71408
$112$	0	0
$113$	−11.0676	−1.04115	−0.520575	−	0.853816i	$-0.674282\pi$
$113$	−11.0676	−1.04115	−0.520575	+	0.853816i	$0.674282\pi$
$114$	0	0
$115$	8.05192	0.750845
$116$	0	0
$117$	−1.66990	−0.154382
$118$	0	0
$119$	8.80108	0.806794
$120$	0	0
$121$	17.7720	1.61563
$122$	0	0
$123$	−14.9665	−1.34949
$124$	0	0
$125$	−11.9887	−1.07230
$126$	0	0
$127$	−0.685828	−0.0608574	−0.0304287	−	0.999537i	$-0.509687\pi$
$127$	−0.685828	−0.0608574	−0.0304287	+	0.999537i	$0.509687\pi$
$128$	0	0
$129$	14.1440	1.24531
$130$	0	0
$131$	−3.84585	−0.336014	−0.168007	−	0.985786i	$-0.553733\pi$
$131$	−3.84585	−0.336014	−0.168007	+	0.985786i	$0.553733\pi$
$132$	0	0
$133$	5.71605	0.495644
$134$	0	0
$135$	5.64890	0.486180
$136$	0	0
$137$	17.9202	1.53103	0.765513	−	0.643420i	$-0.222485\pi$
$137$	17.9202	1.53103	0.765513	+	0.643420i	$0.222485\pi$
$138$	0	0
$139$	10.2754	0.871546	0.435773	−	0.900057i	$-0.356475\pi$
$139$	10.2754	0.871546	0.435773	+	0.900057i	$0.356475\pi$
$140$	0	0
$141$	−9.05103	−0.762234
$142$	0	0
$143$	6.64532	0.555709
$144$	0	0
$145$	−0.415928	−0.0345409
$146$	0	0
$147$	11.5202	0.950169
$148$	0	0
$149$	6.41989	0.525938	0.262969	−	0.964804i	$-0.415298\pi$
$149$	6.41989	0.525938	0.262969	+	0.964804i	$0.415298\pi$
$150$	0	0
$151$	5.33341	0.434027	0.217013	−	0.976169i	$-0.430368\pi$
$151$	5.33341	0.434027	0.217013	+	0.976169i	$0.430368\pi$
$152$	0	0
$153$	9.76733	0.789642
$154$	0	0
$155$	5.96215	0.478892
$156$	0	0
$157$	−9.01684	−0.719622	−0.359811	−	0.933025i	$-0.617159\pi$
$157$	−9.01684	−0.719622	−0.359811	+	0.933025i	$0.617159\pi$
$158$	0	0
$159$	5.05140	0.400602
$160$	0	0
$161$	5.96390	0.470021
$162$	0	0
$163$	17.5918	1.37790	0.688949	−	0.724809i	$-0.258072\pi$
$163$	17.5918	1.37790	0.688949	+	0.724809i	$0.258072\pi$
$164$	0	0
$165$	18.3406	1.42782
$166$	0	0
$167$	−11.5554	−0.894183	−0.447092	−	0.894488i	$-0.647540\pi$
$167$	−11.5554	−0.894183	−0.447092	+	0.894488i	$0.647540\pi$
$168$	0	0
$169$	−11.4652	−0.881936
$170$	0	0
$171$	6.34359	0.485107
$172$	0	0
$173$	10.1903	0.774753	0.387377	−	0.921922i	$-0.373381\pi$
$173$	10.1903	0.774753	0.387377	+	0.921922i	$0.373381\pi$
$174$	0	0
$175$	−2.80695	−0.212185
$176$	0	0
$177$	−8.25824	−0.620728
$178$	0	0
$179$	4.63483	0.346423	0.173212	−	0.984885i	$-0.444586\pi$
$179$	4.63483	0.346423	0.173212	+	0.984885i	$0.444586\pi$
$180$	0	0
$181$	−21.3980	−1.59050	−0.795251	−	0.606280i	$-0.792661\pi$
$181$	−21.3980	−1.59050	−0.795251	+	0.606280i	$0.792661\pi$
$182$	0	0
$183$	31.4869	2.32758
$184$	0	0
$185$	−14.2017	−1.04413
$186$	0	0
$187$	−38.8688	−2.84237
$188$	0	0
$189$	4.18403	0.304344
$190$	0	0
$191$	13.0660	0.945420	0.472710	−	0.881218i	$-0.343276\pi$
$191$	13.0660	0.945420	0.472710	+	0.881218i	$0.343276\pi$
$192$	0	0
$193$	13.9757	1.00599	0.502997	−	0.864288i	$-0.332231\pi$
$193$	13.9757	1.00599	0.502997	+	0.864288i	$0.332231\pi$
$194$	0	0
$195$	4.23604	0.303349
$196$	0	0
$197$	−3.06150	−0.218123	−0.109061	−	0.994035i	$-0.534784\pi$
$197$	−3.06150	−0.218123	−0.109061	+	0.994035i	$0.534784\pi$
$198$	0	0
$199$	−17.1964	−1.21902	−0.609509	−	0.792779i	$-0.708633\pi$
$199$	−17.1964	−1.21902	−0.609509	+	0.792779i	$0.708633\pi$
$200$	0	0
$201$	−15.4372	−1.08886
$202$	0	0
$203$	−0.308070	−0.0216223
$204$	0	0
$205$	11.7698	0.822041
$206$	0	0
$207$	6.61866	0.460028
$208$	0	0
$209$	−25.2441	−1.74617
$210$	0	0
$211$	0.502409	0.0345873	0.0172936	−	0.999850i	$-0.494495\pi$
$211$	0.502409	0.0345873	0.0172936	+	0.999850i	$0.494495\pi$
$212$	0	0
$213$	−22.7423	−1.55828
$214$	0	0
$215$	−11.1230	−0.758582
$216$	0	0
$217$	4.41605	0.299781
$218$	0	0
$219$	13.5546	0.915932
$220$	0	0
$221$	−8.97732	−0.603880
$222$	0	0
$223$	−7.70985	−0.516289	−0.258145	−	0.966106i	$-0.583111\pi$
$223$	−7.70985	−0.516289	−0.258145	+	0.966106i	$0.583111\pi$
$224$	0	0
$225$	−3.11512	−0.207674
$226$	0	0
$227$	−27.6825	−1.83735	−0.918677	−	0.395010i	$-0.870741\pi$
$227$	−27.6825	−1.83735	−0.918677	+	0.395010i	$0.870741\pi$
$228$	0	0
$229$	19.5149	1.28958	0.644789	−	0.764360i	$-0.276945\pi$
$229$	19.5149	1.28958	0.644789	+	0.764360i	$0.276945\pi$
$230$	0	0
$231$	13.5845	0.893797
$232$	0	0
$233$	−16.3435	−1.07070	−0.535350	−	0.844630i	$-0.679820\pi$
$233$	−16.3435	−1.07070	−0.535350	+	0.844630i	$0.679820\pi$
$234$	0	0
$235$	7.11782	0.464315
$236$	0	0
$237$	−11.6826	−0.758868
$238$	0	0
$239$	27.8071	1.79869	0.899346	−	0.437237i	$-0.144043\pi$
$239$	27.8071	1.79869	0.899346	+	0.437237i	$0.144043\pi$
$240$	0	0
$241$	24.7681	1.59545	0.797726	−	0.603020i	$-0.206036\pi$
$241$	24.7681	1.59545	0.797726	+	0.603020i	$0.206036\pi$
$242$	0	0
$243$	13.0752	0.838774
$244$	0	0
$245$	−9.05959	−0.578796
$246$	0	0
$247$	−5.83051	−0.370986
$248$	0	0
$249$	10.0831	0.638990
$250$	0	0
$251$	21.8114	1.37672	0.688361	−	0.725369i	$-0.258331\pi$
$251$	21.8114	1.37672	0.688361	+	0.725369i	$0.258331\pi$
$252$	0	0
$253$	−26.3387	−1.65590
$254$	0	0
$255$	−24.7768	−1.55158
$256$	0	0
$257$	20.0075	1.24803	0.624016	−	0.781412i	$-0.285500\pi$
$257$	20.0075	1.24803	0.624016	+	0.781412i	$0.285500\pi$
$258$	0	0
$259$	−10.5190	−0.653616
$260$	0	0
$261$	−0.341892	−0.0211626
$262$	0	0
$263$	−26.2300	−1.61741	−0.808704	−	0.588215i	$-0.799831\pi$
$263$	−26.2300	−1.61741	−0.808704	+	0.588215i	$0.799831\pi$
$264$	0	0
$265$	−3.97247	−0.244027
$266$	0	0
$267$	−31.5794	−1.93263
$268$	0	0
$269$	20.9217	1.27562	0.637809	−	0.770194i	$-0.279841\pi$
$269$	20.9217	1.27562	0.637809	+	0.770194i	$0.279841\pi$
$270$	0	0
$271$	−7.97979	−0.484738	−0.242369	−	0.970184i	$-0.577924\pi$
$271$	−7.97979	−0.484738	−0.242369	+	0.970184i	$0.577924\pi$
$272$	0	0
$273$	3.13755	0.189893
$274$	0	0
$275$	12.3965	0.747537
$276$	0	0
$277$	27.3326	1.64226	0.821129	−	0.570743i	$-0.193345\pi$
$277$	27.3326	1.64226	0.821129	+	0.570743i	$0.193345\pi$
$278$	0	0
$279$	4.90088	0.293408
$280$	0	0
$281$	8.60346	0.513239	0.256620	−	0.966512i	$-0.417391\pi$
$281$	8.60346	0.513239	0.256620	+	0.966512i	$0.417391\pi$
$282$	0	0
$283$	14.9804	0.890495	0.445247	−	0.895408i	$-0.353116\pi$
$283$	14.9804	0.890495	0.445247	+	0.895408i	$0.353116\pi$
$284$	0	0
$285$	−16.0918	−0.953197
$286$	0	0
$287$	8.71769	0.514589
$288$	0	0
$289$	35.5088	2.08875
$290$	0	0
$291$	−7.33751	−0.430132
$292$	0	0
$293$	−12.1714	−0.711063	−0.355532	−	0.934664i	$-0.615700\pi$
$293$	−12.1714	−0.711063	−0.355532	+	0.934664i	$0.615700\pi$
$294$	0	0
$295$	6.49437	0.378117
$296$	0	0
$297$	−18.4782	−1.07221
$298$	0	0
$299$	−6.08333	−0.351808
$300$	0	0
$301$	−8.23859	−0.474864
$302$	0	0
$303$	23.6261	1.35728
$304$	0	0
$305$	−24.7616	−1.41785
$306$	0	0
$307$	29.0368	1.65722	0.828608	−	0.559830i	$-0.189133\pi$
$307$	29.0368	1.65722	0.828608	+	0.559830i	$0.189133\pi$
$308$	0	0
$309$	0.615008	0.0349866
$310$	0	0
$311$	−31.0007	−1.75789	−0.878945	−	0.476923i	$-0.841752\pi$
$311$	−31.0007	−1.75789	−0.878945	+	0.476923i	$0.841752\pi$
$312$	0	0
$313$	16.3998	0.926969	0.463484	−	0.886105i	$-0.346599\pi$
$313$	16.3998	0.926969	0.463484	+	0.886105i	$0.346599\pi$
$314$	0	0
$315$	2.68454	0.151256
$316$	0	0
$317$	−25.5102	−1.43280	−0.716399	−	0.697691i	$-0.754211\pi$
$317$	−25.5102	−1.43280	−0.716399	+	0.697691i	$0.754211\pi$
$318$	0	0
$319$	1.36055	0.0761760
$320$	0	0
$321$	−27.3619	−1.52719
$322$	0	0
$323$	34.1029	1.89754
$324$	0	0
$325$	2.86316	0.158819
$326$	0	0
$327$	9.44940	0.522553
$328$	0	0
$329$	5.27203	0.290656
$330$	0	0
$331$	16.1883	0.889791	0.444895	−	0.895583i	$-0.353241\pi$
$331$	16.1883	0.889791	0.444895	+	0.895583i	$0.353241\pi$
$332$	0	0
$333$	−11.6738	−0.639720
$334$	0	0
$335$	12.1400	0.663279
$336$	0	0
$337$	16.4117	0.894004	0.447002	−	0.894533i	$-0.352492\pi$
$337$	16.4117	0.894004	0.447002	+	0.894533i	$0.352492\pi$
$338$	0	0
$339$	23.0777	1.25341
$340$	0	0
$341$	−19.5029	−1.05614
$342$	0	0
$343$	−15.2122	−0.821382
$344$	0	0
$345$	−16.7896	−0.903920
$346$	0	0
$347$	8.41921	0.451967	0.225983	−	0.974131i	$-0.427441\pi$
$347$	8.41921	0.451967	0.225983	+	0.974131i	$0.427441\pi$
$348$	0	0
$349$	33.8602	1.81249	0.906247	−	0.422750i	$-0.138935\pi$
$349$	33.8602	1.81249	0.906247	+	0.422750i	$0.138935\pi$
$350$	0	0
$351$	−4.26782	−0.227799
$352$	0	0
$353$	−31.7933	−1.69219	−0.846094	−	0.533033i	$-0.821052\pi$
$353$	−31.7933	−1.69219	−0.846094	+	0.533033i	$0.821052\pi$
$354$	0	0
$355$	17.8848	0.949227
$356$	0	0
$357$	−18.3517	−0.971275
$358$	0	0
$359$	−0.298378	−0.0157478	−0.00787389	−	0.999969i	$-0.502506\pi$
$359$	−0.298378	−0.0157478	−0.00787389	+	0.999969i	$0.502506\pi$
$360$	0	0
$361$	3.14885	0.165729
$362$	0	0
$363$	−37.0575	−1.94501
$364$	0	0
$365$	−10.6594	−0.557941
$366$	0	0
$367$	19.9665	1.04224	0.521122	−	0.853482i	$-0.325513\pi$
$367$	19.9665	1.04224	0.521122	+	0.853482i	$0.325513\pi$
$368$	0	0
$369$	9.67477	0.503649
$370$	0	0
$371$	−2.94233	−0.152758
$372$	0	0
$373$	−1.06482	−0.0551341	−0.0275670	−	0.999620i	$-0.508776\pi$
$373$	−1.06482	−0.0551341	−0.0275670	+	0.999620i	$0.508776\pi$
$374$	0	0
$375$	24.9983	1.29091
$376$	0	0
$377$	0.314239	0.0161841
$378$	0	0
$379$	18.5876	0.954783	0.477392	−	0.878691i	$-0.341582\pi$
$379$	18.5876	0.954783	0.477392	+	0.878691i	$0.341582\pi$
$380$	0	0
$381$	1.43006	0.0732644
$382$	0	0
$383$	1.81749	0.0928694	0.0464347	−	0.998921i	$-0.485214\pi$
$383$	1.81749	0.0928694	0.0464347	+	0.998921i	$0.485214\pi$
$384$	0	0
$385$	−10.6830	−0.544457
$386$	0	0
$387$	−9.14308	−0.464769
$388$	0	0
$389$	−22.5293	−1.14228	−0.571139	−	0.820853i	$-0.693498\pi$
$389$	−22.5293	−1.14228	−0.571139	+	0.820853i	$0.693498\pi$
$390$	0	0
$391$	35.5817	1.79944
$392$	0	0
$393$	8.01923	0.404517
$394$	0	0
$395$	9.18733	0.462265
$396$	0	0
$397$	35.4082	1.77709	0.888544	−	0.458791i	$-0.151717\pi$
$397$	35.4082	1.77709	0.888544	+	0.458791i	$0.151717\pi$
$398$	0	0
$399$	−11.9189	−0.596691
$400$	0	0
$401$	−7.05382	−0.352251	−0.176125	−	0.984368i	$-0.556356\pi$
$401$	−7.05382	−0.352251	−0.176125	+	0.984368i	$0.556356\pi$
$402$	0	0
$403$	−4.50448	−0.224384
$404$	0	0
$405$	−18.4097	−0.914788
$406$	0	0
$407$	46.4555	2.30271
$408$	0	0
$409$	−28.7352	−1.42086	−0.710432	−	0.703766i	$-0.751500\pi$
$409$	−28.7352	−1.42086	−0.710432	+	0.703766i	$0.751500\pi$
$410$	0	0
$411$	−37.3666	−1.84316
$412$	0	0
$413$	4.81025	0.236697
$414$	0	0
$415$	−7.92945	−0.389241
$416$	0	0
$417$	−21.4258	−1.04923
$418$	0	0
$419$	−14.3921	−0.703100	−0.351550	−	0.936169i	$-0.614345\pi$
$419$	−14.3921	−0.703100	−0.351550	+	0.936169i	$0.614345\pi$
$420$	0	0
$421$	−34.4002	−1.67656	−0.838282	−	0.545237i	$-0.816440\pi$
$421$	−34.4002	−1.67656	−0.838282	+	0.545237i	$0.816440\pi$
$422$	0	0
$423$	5.85083	0.284477
$424$	0	0
$425$	−16.7467	−0.812336
$426$	0	0
$427$	−18.3404	−0.887556
$428$	0	0
$429$	−13.8566	−0.669002
$430$	0	0
$431$	19.8728	0.957237	0.478619	−	0.878023i	$-0.341138\pi$
$431$	19.8728	0.957237	0.478619	+	0.878023i	$0.341138\pi$
$432$	0	0
$433$	34.3331	1.64994	0.824972	−	0.565174i	$-0.191191\pi$
$433$	34.3331	1.64994	0.824972	+	0.565174i	$0.191191\pi$
$434$	0	0
$435$	0.867278	0.0415828
$436$	0	0
$437$	23.1092	1.10547
$438$	0	0
$439$	17.2781	0.824640	0.412320	−	0.911039i	$-0.364719\pi$
$439$	17.2781	0.824640	0.412320	+	0.911039i	$0.364719\pi$
$440$	0	0
$441$	−7.44696	−0.354617
$442$	0	0
$443$	−27.2634	−1.29532	−0.647661	−	0.761928i	$-0.724253\pi$
$443$	−27.2634	−1.29532	−0.647661	+	0.761928i	$0.724253\pi$
$444$	0	0
$445$	24.8344	1.17726
$446$	0	0
$447$	−13.3865	−0.633160
$448$	0	0
$449$	13.7996	0.651243	0.325621	−	0.945500i	$-0.394427\pi$
$449$	13.7996	0.651243	0.325621	+	0.945500i	$0.394427\pi$
$450$	0	0
$451$	−38.5005	−1.81292
$452$	0	0
$453$	−11.1210	−0.522512
$454$	0	0
$455$	−2.46740	−0.115674
$456$	0	0
$457$	27.1449	1.26978	0.634892	−	0.772601i	$-0.281045\pi$
$457$	27.1449	1.26978	0.634892	+	0.772601i	$0.281045\pi$
$458$	0	0
$459$	24.9627	1.16516
$460$	0	0
$461$	19.1212	0.890565	0.445283	−	0.895390i	$-0.353103\pi$
$461$	19.1212	0.890565	0.445283	+	0.895390i	$0.353103\pi$
$462$	0	0
$463$	6.42123	0.298420	0.149210	−	0.988806i	$-0.452327\pi$
$463$	6.42123	0.298420	0.149210	+	0.988806i	$0.452327\pi$
$464$	0	0
$465$	−12.4321	−0.576523
$466$	0	0
$467$	3.86476	0.178840	0.0894198	−	0.995994i	$-0.471499\pi$
$467$	3.86476	0.178840	0.0894198	+	0.995994i	$0.471499\pi$
$468$	0	0
$469$	8.99187	0.415206
$470$	0	0
$471$	18.8016	0.866331
$472$	0	0
$473$	36.3846	1.67297
$474$	0	0
$475$	−10.8765	−0.499049
$476$	0	0
$477$	−3.26536	−0.149511
$478$	0	0
$479$	−25.0522	−1.14466	−0.572331	−	0.820023i	$-0.693961\pi$
$479$	−25.0522	−1.14466	−0.572331	+	0.820023i	$0.693961\pi$
$480$	0	0
$481$	10.7296	0.489227
$482$	0	0
$483$	−12.4357	−0.565844
$484$	0	0
$485$	5.77029	0.262016
$486$	0	0
$487$	16.6592	0.754902	0.377451	−	0.926030i	$-0.376801\pi$
$487$	16.6592	0.754902	0.377451	+	0.926030i	$0.376801\pi$
$488$	0	0
$489$	−36.6818	−1.65881
$490$	0	0
$491$	16.6774	0.752642	0.376321	−	0.926489i	$-0.377189\pi$
$491$	16.6774	0.752642	0.376321	+	0.926489i	$0.377189\pi$
$492$	0	0
$493$	−1.83800	−0.0827792
$494$	0	0
$495$	−11.8559	−0.532882
$496$	0	0
$497$	13.2469	0.594206
$498$	0	0
$499$	29.1842	1.30647	0.653233	−	0.757157i	$-0.273412\pi$
$499$	29.1842	1.30647	0.653233	+	0.757157i	$0.273412\pi$
$500$	0	0
$501$	24.0949	1.07648
$502$	0	0
$503$	1.17761	0.0525072	0.0262536	−	0.999655i	$-0.491642\pi$
$503$	1.17761	0.0525072	0.0262536	+	0.999655i	$0.491642\pi$
$504$	0	0
$505$	−18.5798	−0.826789
$506$	0	0
$507$	23.9067	1.06174
$508$	0	0
$509$	−19.1660	−0.849519	−0.424760	−	0.905306i	$-0.639641\pi$
$509$	−19.1660	−0.849519	−0.424760	+	0.905306i	$0.639641\pi$
$510$	0	0
$511$	−7.89524	−0.349265
$512$	0	0
$513$	16.2125	0.715800
$514$	0	0
$515$	−0.483649	−0.0213121
$516$	0	0
$517$	−23.2832	−1.02399
$518$	0	0
$519$	−21.2484	−0.932702
$520$	0	0
$521$	12.9672	0.568103	0.284052	−	0.958809i	$-0.408321\pi$
$521$	12.9672	0.568103	0.284052	+	0.958809i	$0.408321\pi$
$522$	0	0
$523$	2.28182	0.0997769	0.0498884	−	0.998755i	$-0.484113\pi$
$523$	2.28182	0.0997769	0.0498884	+	0.998755i	$0.484113\pi$
$524$	0	0
$525$	5.85295	0.255444
$526$	0	0
$527$	26.3469	1.14769
$528$	0	0
$529$	1.11128	0.0483166
$530$	0	0
$531$	5.33835	0.231665
$532$	0	0
$533$	−8.89226	−0.385166
$534$	0	0
$535$	21.5177	0.930292
$536$	0	0
$537$	−9.66438	−0.417049
$538$	0	0
$539$	29.6349	1.27647
$540$	0	0
$541$	43.9203	1.88828	0.944140	−	0.329546i	$-0.106896\pi$
$541$	43.9203	1.88828	0.944140	+	0.329546i	$0.106896\pi$
$542$	0	0
$543$	44.6184	1.91476
$544$	0	0
$545$	−7.43110	−0.318313
$546$	0	0
$547$	−28.2304	−1.20704	−0.603522	−	0.797346i	$-0.706237\pi$
$547$	−28.2304	−1.20704	−0.603522	+	0.797346i	$0.706237\pi$
$548$	0	0
$549$	−20.3540	−0.868687
$550$	0	0
$551$	−1.19373	−0.0508544
$552$	0	0
$553$	6.80488	0.289373
$554$	0	0
$555$	29.6129	1.25700
$556$	0	0
$557$	5.48119	0.232246	0.116123	−	0.993235i	$-0.462953\pi$
$557$	5.48119	0.232246	0.116123	+	0.993235i	$0.462953\pi$
$558$	0	0
$559$	8.40357	0.355433
$560$	0	0
$561$	81.0477	3.42184
$562$	0	0
$563$	−25.8444	−1.08921	−0.544606	−	0.838692i	$-0.683321\pi$
$563$	−25.8444	−1.08921	−0.544606	+	0.838692i	$0.683321\pi$
$564$	0	0
$565$	−18.1485	−0.763514
$566$	0	0
$567$	−13.6357	−0.572647
$568$	0	0
$569$	19.3103	0.809530	0.404765	−	0.914421i	$-0.367353\pi$
$569$	19.3103	0.809530	0.404765	+	0.914421i	$0.367353\pi$
$570$	0	0
$571$	31.5817	1.32165	0.660826	−	0.750539i	$-0.270206\pi$
$571$	31.5817	1.32165	0.660826	+	0.750539i	$0.270206\pi$
$572$	0	0
$573$	−27.2447	−1.13816
$574$	0	0
$575$	−11.3481	−0.473250
$576$	0	0
$577$	−22.2401	−0.925867	−0.462934	−	0.886393i	$-0.653203\pi$
$577$	−22.2401	−0.925867	−0.462934	+	0.886393i	$0.653203\pi$
$578$	0	0
$579$	−29.1417	−1.21109
$580$	0	0
$581$	−5.87319	−0.243661
$582$	0	0
$583$	12.9944	0.538173
$584$	0	0
$585$	−2.73829	−0.113214
$586$	0	0
$587$	26.1451	1.07912	0.539561	−	0.841946i	$-0.318590\pi$
$587$	26.1451	1.07912	0.539561	+	0.841946i	$0.318590\pi$
$588$	0	0
$589$	17.1116	0.705070
$590$	0	0
$591$	6.38372	0.262591
$592$	0	0
$593$	9.88438	0.405903	0.202951	−	0.979189i	$-0.434947\pi$
$593$	9.88438	0.405903	0.202951	+	0.979189i	$0.434947\pi$
$594$	0	0
$595$	14.4320	0.591653
$596$	0	0
$597$	35.8572	1.46754
$598$	0	0
$599$	30.7640	1.25698	0.628492	−	0.777816i	$-0.283672\pi$
$599$	30.7640	1.25698	0.628492	+	0.777816i	$0.283672\pi$
$600$	0	0
$601$	42.9431	1.75169	0.875844	−	0.482595i	$-0.160306\pi$
$601$	42.9431	1.75169	0.875844	+	0.482595i	$0.160306\pi$
$602$	0	0
$603$	9.97905	0.406378
$604$	0	0
$605$	29.1424	1.18481
$606$	0	0
$607$	−20.4684	−0.830786	−0.415393	−	0.909642i	$-0.636356\pi$
$607$	−20.4684	−0.830786	−0.415393	+	0.909642i	$0.636356\pi$
$608$	0	0
$609$	0.642376	0.0260304
$610$	0	0
$611$	−5.37760	−0.217554
$612$	0	0
$613$	16.6659	0.673128	0.336564	−	0.941661i	$-0.390735\pi$
$613$	16.6659	0.673128	0.336564	+	0.941661i	$0.390735\pi$
$614$	0	0
$615$	−24.5420	−0.989630
$616$	0	0
$617$	31.4150	1.26472	0.632359	−	0.774675i	$-0.282087\pi$
$617$	31.4150	1.26472	0.632359	+	0.774675i	$0.282087\pi$
$618$	0	0
$619$	38.7960	1.55934	0.779671	−	0.626189i	$-0.215386\pi$
$619$	38.7960	1.55934	0.779671	+	0.626189i	$0.215386\pi$
$620$	0	0
$621$	16.9155	0.678796
$622$	0	0
$623$	18.3944	0.736954
$624$	0	0
$625$	−8.10354	−0.324142
$626$	0	0
$627$	52.6381	2.10216
$628$	0	0
$629$	−62.7579	−2.50232
$630$	0	0
$631$	−39.7129	−1.58094	−0.790472	−	0.612498i	$-0.790165\pi$
$631$	−39.7129	−1.58094	−0.790472	+	0.612498i	$0.790165\pi$
$632$	0	0
$633$	−1.04761	−0.0416386
$634$	0	0
$635$	−1.12462	−0.0446290
$636$	0	0
$637$	6.84464	0.271194
$638$	0	0
$639$	14.7013	0.581573
$640$	0	0
$641$	−1.99060	−0.0786240	−0.0393120	−	0.999227i	$-0.512517\pi$
$641$	−1.99060	−0.0786240	−0.0393120	+	0.999227i	$0.512517\pi$
$642$	0	0
$643$	10.8733	0.428801	0.214401	−	0.976746i	$-0.431220\pi$
$643$	10.8733	0.428801	0.214401	+	0.976746i	$0.431220\pi$
$644$	0	0
$645$	23.1933	0.913234
$646$	0	0
$647$	−46.6272	−1.83310	−0.916552	−	0.399915i	$-0.869040\pi$
$647$	−46.6272	−1.83310	−0.916552	+	0.399915i	$0.869040\pi$
$648$	0	0
$649$	−21.2438	−0.833892
$650$	0	0
$651$	−9.20819	−0.360897
$652$	0	0
$653$	−10.5993	−0.414784	−0.207392	−	0.978258i	$-0.566498\pi$
$653$	−10.5993	−0.414784	−0.207392	+	0.978258i	$0.566498\pi$
$654$	0	0
$655$	−6.30640	−0.246412
$656$	0	0
$657$	−8.76204	−0.341840
$658$	0	0
$659$	−38.5045	−1.49992	−0.749961	−	0.661482i	$-0.769928\pi$
$659$	−38.5045	−1.49992	−0.749961	+	0.661482i	$0.769928\pi$
$660$	0	0
$661$	14.4270	0.561146	0.280573	−	0.959833i	$-0.409475\pi$
$661$	14.4270	0.561146	0.280573	+	0.959833i	$0.409475\pi$
$662$	0	0
$663$	18.7192	0.726993
$664$	0	0
$665$	9.37314	0.363475
$666$	0	0
$667$	−1.24549	−0.0482254
$668$	0	0
$669$	16.0763	0.621545
$670$	0	0
$671$	80.9980	3.12689
$672$	0	0
$673$	−13.8746	−0.534827	−0.267414	−	0.963582i	$-0.586169\pi$
$673$	−13.8746	−0.534827	−0.267414	+	0.963582i	$0.586169\pi$
$674$	0	0
$675$	−7.96139	−0.306434
$676$	0	0
$677$	42.9094	1.64914	0.824571	−	0.565759i	$-0.191417\pi$
$677$	42.9094	1.64914	0.824571	+	0.565759i	$0.191417\pi$
$678$	0	0
$679$	4.27394	0.164019
$680$	0	0
$681$	57.7226	2.21193
$682$	0	0
$683$	39.9378	1.52818	0.764089	−	0.645110i	$-0.223189\pi$
$683$	39.9378	1.52818	0.764089	+	0.645110i	$0.223189\pi$
$684$	0	0
$685$	29.3854	1.12276
$686$	0	0
$687$	−40.6917	−1.55248
$688$	0	0
$689$	3.00125	0.114339
$690$	0	0
$691$	−43.9442	−1.67172	−0.835858	−	0.548946i	$-0.815029\pi$
$691$	−43.9442	−1.67172	−0.835858	+	0.548946i	$0.815029\pi$
$692$	0	0
$693$	−8.78142	−0.333579
$694$	0	0
$695$	16.8495	0.639138
$696$	0	0
$697$	52.0112	1.97007
$698$	0	0
$699$	34.0789	1.28898
$700$	0	0
$701$	8.77956	0.331599	0.165800	−	0.986159i	$-0.446979\pi$
$701$	8.77956	0.331599	0.165800	+	0.986159i	$0.446979\pi$
$702$	0	0
$703$	−40.7594	−1.53727
$704$	0	0
$705$	−14.8418	−0.558975
$706$	0	0
$707$	−13.7617	−0.517561
$708$	0	0
$709$	7.34979	0.276027	0.138014	−	0.990430i	$-0.455928\pi$
$709$	7.34979	0.276027	0.138014	+	0.990430i	$0.455928\pi$
$710$	0	0
$711$	7.55197	0.283221
$712$	0	0
$713$	17.8535	0.668620
$714$	0	0
$715$	10.8969	0.407523
$716$	0	0
$717$	−57.9824	−2.16539
$718$	0	0
$719$	32.6437	1.21741	0.608703	−	0.793398i	$-0.291690\pi$
$719$	32.6437	1.21741	0.608703	+	0.793398i	$0.291690\pi$
$720$	0	0
$721$	−0.358229	−0.0133412
$722$	0	0
$723$	−51.6455	−1.92072
$724$	0	0
$725$	0.586197	0.0217708
$726$	0	0
$727$	−30.7202	−1.13935	−0.569674	−	0.821871i	$-0.692930\pi$
$727$	−30.7202	−1.13935	−0.569674	+	0.821871i	$0.692930\pi$
$728$	0	0
$729$	6.41668	0.237655
$730$	0	0
$731$	−49.1528	−1.81798
$732$	0	0
$733$	−1.06049	−0.0391701	−0.0195850	−	0.999808i	$-0.506235\pi$
$733$	−1.06049	−0.0391701	−0.0195850	+	0.999808i	$0.506235\pi$
$734$	0	0
$735$	18.8907	0.696795
$736$	0	0
$737$	−39.7113	−1.46279
$738$	0	0
$739$	46.1526	1.69775	0.848875	−	0.528593i	$-0.177280\pi$
$739$	46.1526	1.69775	0.848875	+	0.528593i	$0.177280\pi$
$740$	0	0
$741$	12.1576	0.446619
$742$	0	0
$743$	30.3668	1.11405	0.557025	−	0.830496i	$-0.311943\pi$
$743$	30.3668	1.11405	0.557025	+	0.830496i	$0.311943\pi$
$744$	0	0
$745$	10.5273	0.385690
$746$	0	0
$747$	−6.51798	−0.238481
$748$	0	0
$749$	15.9378	0.582353
$750$	0	0
$751$	−28.3422	−1.03422	−0.517111	−	0.855918i	$-0.672993\pi$
$751$	−28.3422	−1.03422	−0.517111	+	0.855918i	$0.672993\pi$
$752$	0	0
$753$	−45.4803	−1.65739
$754$	0	0
$755$	8.74570	0.318288
$756$	0	0
$757$	32.8616	1.19437	0.597187	−	0.802102i	$-0.296285\pi$
$757$	32.8616	1.19437	0.597187	+	0.802102i	$0.296285\pi$
$758$	0	0
$759$	54.9206	1.99349
$760$	0	0
$761$	6.23992	0.226197	0.113098	−	0.993584i	$-0.463922\pi$
$761$	6.23992	0.226197	0.113098	+	0.993584i	$0.463922\pi$
$762$	0	0
$763$	−5.50407	−0.199261
$764$	0	0
$765$	16.0164	0.579074
$766$	0	0
$767$	−4.90658	−0.177166
$768$	0	0
$769$	43.9152	1.58362	0.791812	−	0.610765i	$-0.209138\pi$
$769$	43.9152	1.58362	0.791812	+	0.610765i	$0.209138\pi$
$770$	0	0
$771$	−41.7188	−1.50247
$772$	0	0
$773$	33.1771	1.19330	0.596648	−	0.802503i	$-0.296499\pi$
$773$	33.1771	1.19330	0.596648	+	0.802503i	$0.296499\pi$
$774$	0	0
$775$	−8.40289	−0.301841
$776$	0	0
$777$	21.9337	0.786868
$778$	0	0
$779$	33.7798	1.21029
$780$	0	0
$781$	−58.5032	−2.09341
$782$	0	0
$783$	−0.873783	−0.0312265
$784$	0	0
$785$	−14.7858	−0.527726
$786$	0	0
$787$	4.29085	0.152952	0.0764761	−	0.997071i	$-0.475633\pi$
$787$	4.29085	0.152952	0.0764761	+	0.997071i	$0.475633\pi$
$788$	0	0
$789$	54.6937	1.94715
$790$	0	0
$791$	−13.4423	−0.477952
$792$	0	0
$793$	18.7077	0.664330
$794$	0	0
$795$	8.28325	0.293777
$796$	0	0
$797$	−35.4909	−1.25715	−0.628576	−	0.777748i	$-0.716362\pi$
$797$	−35.4909	−1.25715	−0.628576	+	0.777748i	$0.716362\pi$
$798$	0	0
$799$	31.4538	1.11276
$800$	0	0
$801$	20.4138	0.721287
$802$	0	0
$803$	34.8682	1.23047
$804$	0	0
$805$	9.77957	0.344684
$806$	0	0
$807$	−43.6252	−1.53568
$808$	0	0
$809$	49.3587	1.73536	0.867679	−	0.497124i	$-0.165611\pi$
$809$	49.3587	1.73536	0.867679	+	0.497124i	$0.165611\pi$
$810$	0	0
$811$	−37.9419	−1.33232	−0.666160	−	0.745809i	$-0.732063\pi$
$811$	−37.9419	−1.33232	−0.666160	+	0.745809i	$0.732063\pi$
$812$	0	0
$813$	16.6392	0.583561
$814$	0	0
$815$	28.8470	1.01047
$816$	0	0
$817$	−31.9233	−1.11686
$818$	0	0
$819$	−2.02820	−0.0708711
$820$	0	0
$821$	−35.6536	−1.24432	−0.622159	−	0.782891i	$-0.713744\pi$
$821$	−35.6536	−1.24432	−0.622159	+	0.782891i	$0.713744\pi$
$822$	0	0
$823$	−18.1192	−0.631596	−0.315798	−	0.948826i	$-0.602272\pi$
$823$	−18.1192	−0.631596	−0.315798	+	0.948826i	$0.602272\pi$
$824$	0	0
$825$	−25.8487	−0.899937
$826$	0	0
$827$	6.22083	0.216320	0.108160	−	0.994134i	$-0.465504\pi$
$827$	6.22083	0.216320	0.108160	+	0.994134i	$0.465504\pi$
$828$	0	0
$829$	−7.06409	−0.245346	−0.122673	−	0.992447i	$-0.539147\pi$
$829$	−7.06409	−0.245346	−0.122673	+	0.992447i	$0.539147\pi$
$830$	0	0
$831$	−56.9930	−1.97706
$832$	0	0
$833$	−40.0346	−1.38712
$834$	0	0
$835$	−18.9485	−0.655739
$836$	0	0
$837$	12.5253	0.432938
$838$	0	0
$839$	0.437173	0.0150929	0.00754645	−	0.999972i	$-0.497598\pi$
$839$	0.437173	0.0150929	0.00754645	+	0.999972i	$0.497598\pi$
$840$	0	0
$841$	−28.9357	−0.997781
$842$	0	0
$843$	−17.9396	−0.617873
$844$	0	0
$845$	−18.8005	−0.646757
$846$	0	0
$847$	21.5852	0.741676
$848$	0	0
$849$	−31.2367	−1.07204
$850$	0	0
$851$	−42.5268	−1.45780
$852$	0	0
$853$	−8.40211	−0.287683	−0.143841	−	0.989601i	$-0.545945\pi$
$853$	−8.40211	−0.287683	−0.143841	+	0.989601i	$0.545945\pi$
$854$	0	0
$855$	10.4022	0.355747
$856$	0	0
$857$	32.6237	1.11440	0.557202	−	0.830377i	$-0.311875\pi$
$857$	32.6237	1.11440	0.557202	+	0.830377i	$0.311875\pi$
$858$	0	0
$859$	−16.6699	−0.568769	−0.284384	−	0.958710i	$-0.591789\pi$
$859$	−16.6699	−0.568769	−0.284384	+	0.958710i	$0.591789\pi$
$860$	0	0
$861$	−18.1778	−0.619498
$862$	0	0
$863$	33.8466	1.15215	0.576075	−	0.817397i	$-0.304583\pi$
$863$	33.8466	1.15215	0.576075	+	0.817397i	$0.304583\pi$
$864$	0	0
$865$	16.7100	0.568156
$866$	0	0
$867$	−74.0416	−2.51459
$868$	0	0
$869$	−30.0528	−1.01947
$870$	0	0
$871$	−9.17193	−0.310779
$872$	0	0
$873$	4.74317	0.160532
$874$	0	0
$875$	−14.5610	−0.492251
$876$	0	0
$877$	29.4648	0.994956	0.497478	−	0.867477i	$-0.334260\pi$
$877$	29.4648	0.994956	0.497478	+	0.867477i	$0.334260\pi$
$878$	0	0
$879$	25.3794	0.856027
$880$	0	0
$881$	−13.2679	−0.447006	−0.223503	−	0.974703i	$-0.571749\pi$
$881$	−13.2679	−0.447006	−0.223503	+	0.974703i	$0.571749\pi$
$882$	0	0
$883$	32.2904	1.08666	0.543329	−	0.839520i	$-0.317164\pi$
$883$	32.2904	1.08666	0.543329	+	0.839520i	$0.317164\pi$
$884$	0	0
$885$	−13.5418	−0.455203
$886$	0	0
$887$	51.8770	1.74186	0.870930	−	0.491407i	$-0.163517\pi$
$887$	51.8770	1.74186	0.870930	+	0.491407i	$0.163517\pi$
$888$	0	0
$889$	−0.832982	−0.0279373
$890$	0	0
$891$	60.2204	2.01746
$892$	0	0
$893$	20.4284	0.683609
$894$	0	0
$895$	7.60016	0.254045
$896$	0	0
$897$	12.6847	0.423531
$898$	0	0
$899$	−0.922238	−0.0307584
$900$	0	0
$901$	−17.5545	−0.584824
$902$	0	0
$903$	17.1788	0.571675
$904$	0	0
$905$	−35.0883	−1.16638
$906$	0	0
$907$	1.88600	0.0626236	0.0313118	−	0.999510i	$-0.490032\pi$
$907$	1.88600	0.0626236	0.0313118	+	0.999510i	$0.490032\pi$
$908$	0	0
$909$	−15.2725	−0.506558
$910$	0	0
$911$	29.4576	0.975974	0.487987	−	0.872851i	$-0.337731\pi$
$911$	29.4576	0.975974	0.487987	+	0.872851i	$0.337731\pi$
$912$	0	0
$913$	25.9381	0.858426
$914$	0	0
$915$	51.6320	1.70690
$916$	0	0
$917$	−4.67103	−0.154251
$918$	0	0
$919$	6.74962	0.222649	0.111325	−	0.993784i	$-0.464491\pi$
$919$	6.74962	0.222649	0.111325	+	0.993784i	$0.464491\pi$
$920$	0	0
$921$	−60.5464	−1.99507
$922$	0	0
$923$	−13.5122	−0.444759
$924$	0	0
$925$	20.0155	0.658106
$926$	0	0
$927$	−0.397558	−0.0130575
$928$	0	0
$929$	−31.1459	−1.02187	−0.510933	−	0.859621i	$-0.670700\pi$
$929$	−31.1459	−1.02187	−0.510933	+	0.859621i	$0.670700\pi$
$930$	0	0
$931$	−26.0013	−0.852158
$932$	0	0
$933$	64.6416	2.11627
$934$	0	0
$935$	−63.7368	−2.08441
$936$	0	0
$937$	−53.1189	−1.73532	−0.867659	−	0.497159i	$-0.834376\pi$
$937$	−53.1189	−1.73532	−0.867659	+	0.497159i	$0.834376\pi$
$938$	0	0
$939$	−34.1962	−1.11595
$940$	0	0
$941$	38.8324	1.26590	0.632951	−	0.774192i	$-0.281844\pi$
$941$	38.8324	1.26590	0.632951	+	0.774192i	$0.281844\pi$
$942$	0	0
$943$	35.2445	1.14772
$944$	0	0
$945$	6.86095	0.223187
$946$	0	0
$947$	−6.39484	−0.207804	−0.103902	−	0.994588i	$-0.533133\pi$
$947$	−6.39484	−0.207804	−0.103902	+	0.994588i	$0.533133\pi$
$948$	0	0
$949$	8.05334	0.261423
$950$	0	0
$951$	53.1930	1.72490
$952$	0	0
$953$	−48.6634	−1.57636	−0.788180	−	0.615444i	$-0.788977\pi$
$953$	−48.6634	−1.57636	−0.788180	+	0.615444i	$0.788977\pi$
$954$	0	0
$955$	21.4255	0.693313
$956$	0	0
$957$	−2.83696	−0.0917060
$958$	0	0
$959$	21.7652	0.702836
$960$	0	0
$961$	−17.7801	−0.573552
$962$	0	0
$963$	17.6875	0.569972
$964$	0	0
$965$	22.9173	0.737734
$966$	0	0
$967$	35.1828	1.13140	0.565702	−	0.824610i	$-0.308605\pi$
$967$	35.1828	1.13140	0.565702	+	0.824610i	$0.308605\pi$
$968$	0	0
$969$	−71.1102	−2.28439
$970$	0	0
$971$	−27.3112	−0.876459	−0.438230	−	0.898863i	$-0.644394\pi$
$971$	−27.3112	−0.876459	−0.438230	+	0.898863i	$0.644394\pi$
$972$	0	0
$973$	12.4801	0.400093
$974$	0	0
$975$	−5.97015	−0.191198
$976$	0	0
$977$	−53.5810	−1.71421	−0.857105	−	0.515142i	$-0.827739\pi$
$977$	−53.5810	−1.71421	−0.857105	+	0.515142i	$0.827739\pi$
$978$	0	0
$979$	−81.2361	−2.59632
$980$	0	0
$981$	−6.10835	−0.195025
$982$	0	0
$983$	61.3246	1.95595	0.977976	−	0.208717i	$-0.0669288\pi$
$983$	61.3246	1.95595	0.977976	+	0.208717i	$0.0669288\pi$
$984$	0	0
$985$	−5.02022	−0.159958
$986$	0	0
$987$	−10.9930	−0.349912
$988$	0	0
$989$	−33.3076	−1.05912
$990$	0	0
$991$	−41.9423	−1.33234	−0.666171	−	0.745799i	$-0.732068\pi$
$991$	−41.9423	−1.33234	−0.666171	+	0.745799i	$0.732068\pi$
$992$	0	0
$993$	−33.7553	−1.07119
$994$	0	0
$995$	−28.1985	−0.893952
$996$	0	0
$997$	−12.8739	−0.407720	−0.203860	−	0.979000i	$-0.565349\pi$
$997$	−12.8739	−0.407720	−0.203860	+	0.979000i	$0.565349\pi$
$998$	0	0
$999$	−29.8351	−0.943940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8012.2.a.b.1.15	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8012.2.a.b.1.15	✓	88	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.08516 q^{3} +1.63979 q^{5} +1.21456 q^{7} +1.34791 q^{9} +O(q^{10})\)	\(q-2.08516 q^{3} +1.63979 q^{5} +1.21456 q^{7} +1.34791 q^{9} -5.36395 q^{11} -1.23888 q^{13} -3.41924 q^{15} +7.24629 q^{17} +4.70626 q^{19} -2.53256 q^{21} +4.91032 q^{23} -2.31108 q^{25} +3.44489 q^{27} -0.253647 q^{29} +3.63592 q^{31} +11.1847 q^{33} +1.99163 q^{35} -8.66068 q^{37} +2.58328 q^{39} +7.17763 q^{41} -6.78317 q^{43} +2.21029 q^{45} +4.34068 q^{47} -5.52484 q^{49} -15.1097 q^{51} -2.42254 q^{53} -8.79577 q^{55} -9.81331 q^{57} +3.96048 q^{59} -15.1004 q^{61} +1.63712 q^{63} -2.03152 q^{65} +7.40337 q^{67} -10.2388 q^{69} +10.9067 q^{71} -6.50048 q^{73} +4.81897 q^{75} -6.51486 q^{77} +5.60274 q^{79} -11.2269 q^{81} -4.83564 q^{83} +11.8824 q^{85} +0.528894 q^{87} +15.1448 q^{89} -1.50470 q^{91} -7.58148 q^{93} +7.71729 q^{95} +3.51891 q^{97} -7.23010 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) \) 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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