LMFDB - Embedded newform 6008.2.a.e.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.684026 q^{3} -2.03205 q^{5} -0.0739397 q^{7} -2.53211 q^{9} +O(q^{10})$	$q-0.684026 q^{3} -2.03205 q^{5} -0.0739397 q^{7} -2.53211 q^{9} -6.59782 q^{11} -2.19215 q^{13} +1.38998 q^{15} -2.82314 q^{17} +3.57602 q^{19} +0.0505766 q^{21} -3.59900 q^{23} -0.870764 q^{25} +3.78411 q^{27} -0.888227 q^{29} -3.39569 q^{31} +4.51308 q^{33} +0.150249 q^{35} -1.90012 q^{37} +1.49949 q^{39} -5.01384 q^{41} -8.72563 q^{43} +5.14538 q^{45} -7.66472 q^{47} -6.99453 q^{49} +1.93110 q^{51} -7.04401 q^{53} +13.4071 q^{55} -2.44609 q^{57} -1.37245 q^{59} +0.987178 q^{61} +0.187223 q^{63} +4.45457 q^{65} -0.134803 q^{67} +2.46181 q^{69} +3.40620 q^{71} +10.5929 q^{73} +0.595625 q^{75} +0.487841 q^{77} -13.9690 q^{79} +5.00790 q^{81} -0.111240 q^{83} +5.73676 q^{85} +0.607570 q^{87} -0.114945 q^{89} +0.162087 q^{91} +2.32274 q^{93} -7.26666 q^{95} -14.8142 q^{97} +16.7064 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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
$3$	−0.684026	−0.394922	−0.197461	−	0.980311i	$-0.563270\pi$
$3$	−0.684026	−0.394922	−0.197461	+	0.980311i	$0.563270\pi$
$4$	0	0
$5$	−2.03205	−0.908761	−0.454381	−	0.890808i	$-0.650139\pi$
$5$	−2.03205	−0.908761	−0.454381	+	0.890808i	$0.650139\pi$
$6$	0	0
$7$	−0.0739397	−0.0279466	−0.0139733	−	0.999902i	$-0.504448\pi$
$7$	−0.0739397	−0.0279466	−0.0139733	+	0.999902i	$0.504448\pi$
$8$	0	0
$9$	−2.53211	−0.844036
$10$	0	0
$11$	−6.59782	−1.98932	−0.994659	−	0.103212i	$-0.967088\pi$
$11$	−6.59782	−1.98932	−0.994659	+	0.103212i	$0.967088\pi$
$12$	0	0
$13$	−2.19215	−0.607994	−0.303997	−	0.952673i	$-0.598321\pi$
$13$	−2.19215	−0.607994	−0.303997	+	0.952673i	$0.598321\pi$
$14$	0	0
$15$	1.38998	0.358890
$16$	0	0
$17$	−2.82314	−0.684711	−0.342356	−	0.939570i	$-0.611225\pi$
$17$	−2.82314	−0.684711	−0.342356	+	0.939570i	$0.611225\pi$
$18$	0	0
$19$	3.57602	0.820395	0.410197	−	0.911997i	$-0.365460\pi$
$19$	3.57602	0.820395	0.410197	+	0.911997i	$0.365460\pi$
$20$	0	0
$21$	0.0505766	0.0110367
$22$	0	0
$23$	−3.59900	−0.750444	−0.375222	−	0.926935i	$-0.622434\pi$
$23$	−3.59900	−0.750444	−0.375222	+	0.926935i	$0.622434\pi$
$24$	0	0
$25$	−0.870764	−0.174153
$26$	0	0
$27$	3.78411	0.728251
$28$	0	0
$29$	−0.888227	−0.164940	−0.0824698	−	0.996594i	$-0.526281\pi$
$29$	−0.888227	−0.164940	−0.0824698	+	0.996594i	$0.526281\pi$
$30$	0	0
$31$	−3.39569	−0.609883	−0.304942	−	0.952371i	$-0.598637\pi$
$31$	−3.39569	−0.609883	−0.304942	+	0.952371i	$0.598637\pi$
$32$	0	0
$33$	4.51308	0.785627
$34$	0	0
$35$	0.150249	0.0253968
$36$	0	0
$37$	−1.90012	−0.312378	−0.156189	−	0.987727i	$-0.549921\pi$
$37$	−1.90012	−0.312378	−0.156189	+	0.987727i	$0.549921\pi$
$38$	0	0
$39$	1.49949	0.240111
$40$	0	0
$41$	−5.01384	−0.783031	−0.391515	−	0.920172i	$-0.628049\pi$
$41$	−5.01384	−0.783031	−0.391515	+	0.920172i	$0.628049\pi$
$42$	0	0
$43$	−8.72563	−1.33065	−0.665323	−	0.746555i	$-0.731706\pi$
$43$	−8.72563	−1.33065	−0.665323	+	0.746555i	$0.731706\pi$
$44$	0	0
$45$	5.14538	0.767028
$46$	0	0
$47$	−7.66472	−1.11801	−0.559007	−	0.829163i	$-0.688818\pi$
$47$	−7.66472	−1.11801	−0.559007	+	0.829163i	$0.688818\pi$
$48$	0	0
$49$	−6.99453	−0.999219
$50$	0	0
$51$	1.93110	0.270408
$52$	0	0
$53$	−7.04401	−0.967570	−0.483785	−	0.875187i	$-0.660738\pi$
$53$	−7.04401	−0.967570	−0.483785	+	0.875187i	$0.660738\pi$
$54$	0	0
$55$	13.4071	1.80782
$56$	0	0
$57$	−2.44609	−0.323992
$58$	0	0
$59$	−1.37245	−0.178677	−0.0893387	−	0.996001i	$-0.528475\pi$
$59$	−1.37245	−0.178677	−0.0893387	+	0.996001i	$0.528475\pi$
$60$	0	0
$61$	0.987178	0.126395	0.0631976	−	0.998001i	$-0.479870\pi$
$61$	0.987178	0.126395	0.0631976	+	0.998001i	$0.479870\pi$
$62$	0	0
$63$	0.187223	0.0235879
$64$	0	0
$65$	4.45457	0.552522
$66$	0	0
$67$	−0.134803	−0.0164687	−0.00823437	−	0.999966i	$-0.502621\pi$
$67$	−0.134803	−0.0164687	−0.00823437	+	0.999966i	$0.502621\pi$
$68$	0	0
$69$	2.46181	0.296367
$70$	0	0
$71$	3.40620	0.404241	0.202121	−	0.979361i	$-0.435217\pi$
$71$	3.40620	0.404241	0.202121	+	0.979361i	$0.435217\pi$
$72$	0	0
$73$	10.5929	1.23980	0.619901	−	0.784680i	$-0.287173\pi$
$73$	10.5929	1.23980	0.619901	+	0.784680i	$0.287173\pi$
$74$	0	0
$75$	0.595625	0.0687768
$76$	0	0
$77$	0.487841	0.0555946
$78$	0	0
$79$	−13.9690	−1.57163	−0.785816	−	0.618460i	$-0.787757\pi$
$79$	−13.9690	−1.57163	−0.785816	+	0.618460i	$0.787757\pi$
$80$	0	0
$81$	5.00790	0.556433
$82$	0	0
$83$	−0.111240	−0.0122102	−0.00610510	−	0.999981i	$-0.501943\pi$
$83$	−0.111240	−0.0122102	−0.00610510	+	0.999981i	$0.501943\pi$
$84$	0	0
$85$	5.73676	0.622239
$86$	0	0
$87$	0.607570	0.0651384
$88$	0	0
$89$	−0.114945	−0.0121841	−0.00609206	−	0.999981i	$-0.501939\pi$
$89$	−0.114945	−0.0121841	−0.00609206	+	0.999981i	$0.501939\pi$
$90$	0	0
$91$	0.162087	0.0169914
$92$	0	0
$93$	2.32274	0.240857
$94$	0	0
$95$	−7.26666	−0.745543
$96$	0	0
$97$	−14.8142	−1.50416	−0.752078	−	0.659074i	$-0.770948\pi$
$97$	−14.8142	−1.50416	−0.752078	+	0.659074i	$0.770948\pi$
$98$	0	0
$99$	16.7064	1.67906
$100$	0	0
$101$	−9.10734	−0.906214	−0.453107	−	0.891456i	$-0.649685\pi$
$101$	−9.10734	−0.906214	−0.453107	+	0.891456i	$0.649685\pi$
$102$	0	0
$103$	−4.35467	−0.429078	−0.214539	−	0.976715i	$-0.568825\pi$
$103$	−4.35467	−0.429078	−0.214539	+	0.976715i	$0.568825\pi$
$104$	0	0
$105$	−0.102774	−0.0100298
$106$	0	0
$107$	12.0099	1.16104	0.580520	−	0.814246i	$-0.302849\pi$
$107$	12.0099	1.16104	0.580520	+	0.814246i	$0.302849\pi$
$108$	0	0
$109$	6.48899	0.621533	0.310766	−	0.950486i	$-0.399414\pi$
$109$	6.48899	0.621533	0.310766	+	0.950486i	$0.399414\pi$
$110$	0	0
$111$	1.29973	0.123365
$112$	0	0
$113$	3.15578	0.296870	0.148435	−	0.988922i	$-0.452576\pi$
$113$	3.15578	0.296870	0.148435	+	0.988922i	$0.452576\pi$
$114$	0	0
$115$	7.31336	0.681975
$116$	0	0
$117$	5.55077	0.513169
$118$	0	0
$119$	0.208742	0.0191353
$120$	0	0
$121$	32.5313	2.95739
$122$	0	0
$123$	3.42960	0.309236
$124$	0	0
$125$	11.9297	1.06702
$126$	0	0
$127$	20.5968	1.82767	0.913834	−	0.406087i	$-0.133107\pi$
$127$	20.5968	1.82767	0.913834	+	0.406087i	$0.133107\pi$
$128$	0	0
$129$	5.96856	0.525502
$130$	0	0
$131$	−9.61578	−0.840134	−0.420067	−	0.907493i	$-0.637994\pi$
$131$	−9.61578	−0.840134	−0.420067	+	0.907493i	$0.637994\pi$
$132$	0	0
$133$	−0.264410	−0.0229272
$134$	0	0
$135$	−7.68950	−0.661807
$136$	0	0
$137$	−13.9773	−1.19416	−0.597080	−	0.802182i	$-0.703673\pi$
$137$	−13.9773	−1.19416	−0.597080	+	0.802182i	$0.703673\pi$
$138$	0	0
$139$	−10.4785	−0.888778	−0.444389	−	0.895834i	$-0.646579\pi$
$139$	−10.4785	−0.888778	−0.444389	+	0.895834i	$0.646579\pi$
$140$	0	0
$141$	5.24287	0.441529
$142$	0	0
$143$	14.4634	1.20949
$144$	0	0
$145$	1.80492	0.149891
$146$	0	0
$147$	4.78444	0.394614
$148$	0	0
$149$	10.2924	0.843186	0.421593	−	0.906785i	$-0.361471\pi$
$149$	10.2924	0.843186	0.421593	+	0.906785i	$0.361471\pi$
$150$	0	0
$151$	11.1865	0.910348	0.455174	−	0.890403i	$-0.349577\pi$
$151$	11.1865	0.910348	0.455174	+	0.890403i	$0.349577\pi$
$152$	0	0
$153$	7.14849	0.577921
$154$	0	0
$155$	6.90021	0.554238
$156$	0	0
$157$	24.2198	1.93295	0.966476	−	0.256756i	$-0.0826537\pi$
$157$	24.2198	1.93295	0.966476	+	0.256756i	$0.0826537\pi$
$158$	0	0
$159$	4.81829	0.382115
$160$	0	0
$161$	0.266109	0.0209723
$162$	0	0
$163$	5.13680	0.402345	0.201172	−	0.979556i	$-0.435525\pi$
$163$	5.13680	0.402345	0.201172	+	0.979556i	$0.435525\pi$
$164$	0	0
$165$	−9.17082	−0.713947
$166$	0	0
$167$	−21.0597	−1.62965	−0.814823	−	0.579709i	$-0.803166\pi$
$167$	−21.0597	−1.62965	−0.814823	+	0.579709i	$0.803166\pi$
$168$	0	0
$169$	−8.19446	−0.630343
$170$	0	0
$171$	−9.05487	−0.692443
$172$	0	0
$173$	−1.32914	−0.101052	−0.0505262	−	0.998723i	$-0.516090\pi$
$173$	−1.32914	−0.101052	−0.0505262	+	0.998723i	$0.516090\pi$
$174$	0	0
$175$	0.0643840	0.00486697
$176$	0	0
$177$	0.938789	0.0705637
$178$	0	0
$179$	2.45619	0.183585	0.0917923	−	0.995778i	$-0.470740\pi$
$179$	2.45619	0.183585	0.0917923	+	0.995778i	$0.470740\pi$
$180$	0	0
$181$	−1.46195	−0.108666	−0.0543329	−	0.998523i	$-0.517303\pi$
$181$	−1.46195	−0.108666	−0.0543329	+	0.998523i	$0.517303\pi$
$182$	0	0
$183$	−0.675255	−0.0499163
$184$	0	0
$185$	3.86114	0.283877
$186$	0	0
$187$	18.6266	1.36211
$188$	0	0
$189$	−0.279795	−0.0203521
$190$	0	0
$191$	−16.9734	−1.22815	−0.614077	−	0.789246i	$-0.710471\pi$
$191$	−16.9734	−1.22815	−0.614077	+	0.789246i	$0.710471\pi$
$192$	0	0
$193$	13.0514	0.939460	0.469730	−	0.882810i	$-0.344351\pi$
$193$	13.0514	0.939460	0.469730	+	0.882810i	$0.344351\pi$
$194$	0	0
$195$	−3.04704	−0.218203
$196$	0	0
$197$	−10.7913	−0.768846	−0.384423	−	0.923157i	$-0.625600\pi$
$197$	−10.7913	−0.768846	−0.384423	+	0.923157i	$0.625600\pi$
$198$	0	0
$199$	−13.1948	−0.935356	−0.467678	−	0.883899i	$-0.654909\pi$
$199$	−13.1948	−0.935356	−0.467678	+	0.883899i	$0.654909\pi$
$200$	0	0
$201$	0.0922084	0.00650388
$202$	0	0
$203$	0.0656752	0.00460950
$204$	0	0
$205$	10.1884	0.711588
$206$	0	0
$207$	9.11307	0.633402
$208$	0	0
$209$	−23.5939	−1.63203
$210$	0	0
$211$	−5.40060	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$211$	−5.40060	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$212$	0	0
$213$	−2.32993	−0.159644
$214$	0	0
$215$	17.7309	1.20924
$216$	0	0
$217$	0.251076	0.0170441
$218$	0	0
$219$	−7.24580	−0.489625
$220$	0	0
$221$	6.18875	0.416301
$222$	0	0
$223$	−11.8012	−0.790267	−0.395133	−	0.918624i	$-0.629302\pi$
$223$	−11.8012	−0.790267	−0.395133	+	0.918624i	$0.629302\pi$
$224$	0	0
$225$	2.20487	0.146991
$226$	0	0
$227$	−4.74047	−0.314636	−0.157318	−	0.987548i	$-0.550285\pi$
$227$	−4.74047	−0.314636	−0.157318	+	0.987548i	$0.550285\pi$
$228$	0	0
$229$	−8.72948	−0.576860	−0.288430	−	0.957501i	$-0.593133\pi$
$229$	−8.72948	−0.576860	−0.288430	+	0.957501i	$0.593133\pi$
$230$	0	0
$231$	−0.333696	−0.0219556
$232$	0	0
$233$	12.1653	0.796978	0.398489	−	0.917173i	$-0.369535\pi$
$233$	12.1653	0.796978	0.398489	+	0.917173i	$0.369535\pi$
$234$	0	0
$235$	15.5751	1.01601
$236$	0	0
$237$	9.55514	0.620673
$238$	0	0
$239$	−19.0737	−1.23377	−0.616886	−	0.787053i	$-0.711606\pi$
$239$	−19.0737	−1.23377	−0.616886	+	0.787053i	$0.711606\pi$
$240$	0	0
$241$	22.1823	1.42889	0.714445	−	0.699692i	$-0.246679\pi$
$241$	22.1823	1.42889	0.714445	+	0.699692i	$0.246679\pi$
$242$	0	0
$243$	−14.7778	−0.947999
$244$	0	0
$245$	14.2133	0.908052
$246$	0	0
$247$	−7.83918	−0.498795
$248$	0	0
$249$	0.0760912	0.00482208
$250$	0	0
$251$	9.02726	0.569795	0.284898	−	0.958558i	$-0.408040\pi$
$251$	9.02726	0.569795	0.284898	+	0.958558i	$0.408040\pi$
$252$	0	0
$253$	23.7456	1.49287
$254$	0	0
$255$	−3.92409	−0.245736
$256$	0	0
$257$	21.5454	1.34396	0.671982	−	0.740567i	$-0.265443\pi$
$257$	21.5454	1.34396	0.671982	+	0.740567i	$0.265443\pi$
$258$	0	0
$259$	0.140494	0.00872988
$260$	0	0
$261$	2.24909	0.139215
$262$	0	0
$263$	−4.45476	−0.274692	−0.137346	−	0.990523i	$-0.543857\pi$
$263$	−4.45476	−0.274692	−0.137346	+	0.990523i	$0.543857\pi$
$264$	0	0
$265$	14.3138	0.879290
$266$	0	0
$267$	0.0786252	0.00481179
$268$	0	0
$269$	8.86029	0.540221	0.270111	−	0.962829i	$-0.412940\pi$
$269$	8.86029	0.540221	0.270111	+	0.962829i	$0.412940\pi$
$270$	0	0
$271$	15.2266	0.924950	0.462475	−	0.886632i	$-0.346961\pi$
$271$	15.2266	0.924950	0.462475	+	0.886632i	$0.346961\pi$
$272$	0	0
$273$	−0.110872	−0.00671027
$274$	0	0
$275$	5.74515	0.346445
$276$	0	0
$277$	−7.58816	−0.455928	−0.227964	−	0.973670i	$-0.573207\pi$
$277$	−7.58816	−0.455928	−0.227964	+	0.973670i	$0.573207\pi$
$278$	0	0
$279$	8.59824	0.514763
$280$	0	0
$281$	−27.6883	−1.65174	−0.825871	−	0.563859i	$-0.809316\pi$
$281$	−27.6883	−1.65174	−0.825871	+	0.563859i	$0.809316\pi$
$282$	0	0
$283$	−2.19348	−0.130389	−0.0651945	−	0.997873i	$-0.520767\pi$
$283$	−2.19348	−0.130389	−0.0651945	+	0.997873i	$0.520767\pi$
$284$	0	0
$285$	4.97058	0.294432
$286$	0	0
$287$	0.370722	0.0218830
$288$	0	0
$289$	−9.02990	−0.531170
$290$	0	0
$291$	10.1333	0.594025
$292$	0	0
$293$	−24.3560	−1.42289	−0.711445	−	0.702742i	$-0.751959\pi$
$293$	−24.3560	−1.42289	−0.711445	+	0.702742i	$0.751959\pi$
$294$	0	0
$295$	2.78888	0.162375
$296$	0	0
$297$	−24.9669	−1.44872
$298$	0	0
$299$	7.88957	0.456266
$300$	0	0
$301$	0.645170	0.0371870
$302$	0	0
$303$	6.22966	0.357884
$304$	0	0
$305$	−2.00600	−0.114863
$306$	0	0
$307$	−10.7080	−0.611137	−0.305569	−	0.952170i	$-0.598847\pi$
$307$	−10.7080	−0.611137	−0.305569	+	0.952170i	$0.598847\pi$
$308$	0	0
$309$	2.97871	0.169453
$310$	0	0
$311$	14.4226	0.817832	0.408916	−	0.912572i	$-0.365907\pi$
$311$	14.4226	0.817832	0.408916	+	0.912572i	$0.365907\pi$
$312$	0	0
$313$	4.86957	0.275244	0.137622	−	0.990485i	$-0.456054\pi$
$313$	4.86957	0.275244	0.137622	+	0.990485i	$0.456054\pi$
$314$	0	0
$315$	−0.380447	−0.0214358
$316$	0	0
$317$	13.9059	0.781032	0.390516	−	0.920596i	$-0.372297\pi$
$317$	13.9059	0.781032	0.390516	+	0.920596i	$0.372297\pi$
$318$	0	0
$319$	5.86037	0.328118
$320$	0	0
$321$	−8.21508	−0.458521
$322$	0	0
$323$	−10.0956	−0.561734
$324$	0	0
$325$	1.90885	0.105884
$326$	0	0
$327$	−4.43864	−0.245457
$328$	0	0
$329$	0.566727	0.0312447
$330$	0	0
$331$	3.63288	0.199681	0.0998405	−	0.995003i	$-0.468167\pi$
$331$	3.63288	0.199681	0.0998405	+	0.995003i	$0.468167\pi$
$332$	0	0
$333$	4.81131	0.263658
$334$	0	0
$335$	0.273926	0.0149662
$336$	0	0
$337$	−17.6872	−0.963485	−0.481742	−	0.876313i	$-0.659996\pi$
$337$	−17.6872	−0.963485	−0.481742	+	0.876313i	$0.659996\pi$
$338$	0	0
$339$	−2.15863	−0.117241
$340$	0	0
$341$	22.4041	1.21325
$342$	0	0
$343$	1.03475	0.0558713
$344$	0	0
$345$	−5.00253	−0.269327
$346$	0	0
$347$	27.7350	1.48889	0.744447	−	0.667682i	$-0.232713\pi$
$347$	27.7350	1.48889	0.744447	+	0.667682i	$0.232713\pi$
$348$	0	0
$349$	18.1023	0.968994	0.484497	−	0.874793i	$-0.339003\pi$
$349$	18.1023	0.968994	0.484497	+	0.874793i	$0.339003\pi$
$350$	0	0
$351$	−8.29534	−0.442773
$352$	0	0
$353$	−26.5805	−1.41474	−0.707368	−	0.706845i	$-0.750118\pi$
$353$	−26.5805	−1.41474	−0.707368	+	0.706845i	$0.750118\pi$
$354$	0	0
$355$	−6.92157	−0.367359
$356$	0	0
$357$	−0.142785	−0.00755697
$358$	0	0
$359$	6.79611	0.358685	0.179343	−	0.983787i	$-0.442603\pi$
$359$	6.79611	0.358685	0.179343	+	0.983787i	$0.442603\pi$
$360$	0	0
$361$	−6.21209	−0.326952
$362$	0	0
$363$	−22.2522	−1.16794
$364$	0	0
$365$	−21.5253	−1.12668
$366$	0	0
$367$	−27.8948	−1.45610	−0.728048	−	0.685526i	$-0.759572\pi$
$367$	−27.8948	−1.45610	−0.728048	+	0.685526i	$0.759572\pi$
$368$	0	0
$369$	12.6956	0.660906
$370$	0	0
$371$	0.520832	0.0270403
$372$	0	0
$373$	34.0340	1.76222	0.881108	−	0.472916i	$-0.156799\pi$
$373$	34.0340	1.76222	0.881108	+	0.472916i	$0.156799\pi$
$374$	0	0
$375$	−8.16022	−0.421392
$376$	0	0
$377$	1.94713	0.100282
$378$	0	0
$379$	19.5596	1.00471	0.502353	−	0.864662i	$-0.332468\pi$
$379$	19.5596	1.00471	0.502353	+	0.864662i	$0.332468\pi$
$380$	0	0
$381$	−14.0887	−0.721787
$382$	0	0
$383$	−32.7659	−1.67426	−0.837131	−	0.547003i	$-0.815769\pi$
$383$	−32.7659	−1.67426	−0.837131	+	0.547003i	$0.815769\pi$
$384$	0	0
$385$	−0.991318	−0.0505223
$386$	0	0
$387$	22.0943	1.12311
$388$	0	0
$389$	0.197018	0.00998919	0.00499459	−	0.999988i	$-0.498410\pi$
$389$	0.197018	0.00998919	0.00499459	+	0.999988i	$0.498410\pi$
$390$	0	0
$391$	10.1605	0.513838
$392$	0	0
$393$	6.57744	0.331788
$394$	0	0
$395$	28.3857	1.42824
$396$	0	0
$397$	4.33378	0.217506	0.108753	−	0.994069i	$-0.465314\pi$
$397$	4.33378	0.217506	0.108753	+	0.994069i	$0.465314\pi$
$398$	0	0
$399$	0.180863	0.00905448
$400$	0	0
$401$	24.6346	1.23020	0.615098	−	0.788451i	$-0.289117\pi$
$401$	24.6346	1.23020	0.615098	+	0.788451i	$0.289117\pi$
$402$	0	0
$403$	7.44387	0.370805
$404$	0	0
$405$	−10.1763	−0.505665
$406$	0	0
$407$	12.5367	0.621419
$408$	0	0
$409$	14.7034	0.727035	0.363518	−	0.931587i	$-0.381576\pi$
$409$	14.7034	0.727035	0.363518	+	0.931587i	$0.381576\pi$
$410$	0	0
$411$	9.56082	0.471601
$412$	0	0
$413$	0.101478	0.00499342
$414$	0	0
$415$	0.226046	0.0110962
$416$	0	0
$417$	7.16759	0.350998
$418$	0	0
$419$	−16.7968	−0.820578	−0.410289	−	0.911955i	$-0.634572\pi$
$419$	−16.7968	−0.820578	−0.410289	+	0.911955i	$0.634572\pi$
$420$	0	0
$421$	27.7889	1.35435	0.677174	−	0.735823i	$-0.263204\pi$
$421$	27.7889	1.35435	0.677174	+	0.735823i	$0.263204\pi$
$422$	0	0
$423$	19.4079	0.943645
$424$	0	0
$425$	2.45829	0.119244
$426$	0	0
$427$	−0.0729916	−0.00353231
$428$	0	0
$429$	−9.89337	−0.477656
$430$	0	0
$431$	−34.2474	−1.64964	−0.824818	−	0.565398i	$-0.808723\pi$
$431$	−34.2474	−1.64964	−0.824818	+	0.565398i	$0.808723\pi$
$432$	0	0
$433$	8.03722	0.386244	0.193122	−	0.981175i	$-0.438139\pi$
$433$	8.03722	0.386244	0.193122	+	0.981175i	$0.438139\pi$
$434$	0	0
$435$	−1.23461	−0.0591952
$436$	0	0
$437$	−12.8701	−0.615661
$438$	0	0
$439$	−9.23717	−0.440866	−0.220433	−	0.975402i	$-0.570747\pi$
$439$	−9.23717	−0.440866	−0.220433	+	0.975402i	$0.570747\pi$
$440$	0	0
$441$	17.7109	0.843377
$442$	0	0
$443$	−37.8221	−1.79698	−0.898490	−	0.438994i	$-0.855335\pi$
$443$	−37.8221	−1.79698	−0.898490	+	0.438994i	$0.855335\pi$
$444$	0	0
$445$	0.233574	0.0110725
$446$	0	0
$447$	−7.04026	−0.332993
$448$	0	0
$449$	−3.04929	−0.143905	−0.0719524	−	0.997408i	$-0.522923\pi$
$449$	−3.04929	−0.143905	−0.0719524	+	0.997408i	$0.522923\pi$
$450$	0	0
$451$	33.0805	1.55770
$452$	0	0
$453$	−7.65188	−0.359517
$454$	0	0
$455$	−0.329370	−0.0154411
$456$	0	0
$457$	−0.581707	−0.0272111	−0.0136056	−	0.999907i	$-0.504331\pi$
$457$	−0.581707	−0.0272111	−0.0136056	+	0.999907i	$0.504331\pi$
$458$	0	0
$459$	−10.6830	−0.498642
$460$	0	0
$461$	10.1282	0.471718	0.235859	−	0.971787i	$-0.424210\pi$
$461$	10.1282	0.471718	0.235859	+	0.971787i	$0.424210\pi$
$462$	0	0
$463$	−8.72104	−0.405301	−0.202651	−	0.979251i	$-0.564956\pi$
$463$	−8.72104	−0.405301	−0.202651	+	0.979251i	$0.564956\pi$
$464$	0	0
$465$	−4.71992	−0.218881
$466$	0	0
$467$	−16.2772	−0.753217	−0.376609	−	0.926372i	$-0.622910\pi$
$467$	−16.2772	−0.753217	−0.376609	+	0.926372i	$0.622910\pi$
$468$	0	0
$469$	0.00996725	0.000460245 0
$470$	0	0
$471$	−16.5670	−0.763366
$472$	0	0
$473$	57.5702	2.64708
$474$	0	0
$475$	−3.11387	−0.142874
$476$	0	0
$477$	17.8362	0.816664
$478$	0	0
$479$	−15.6527	−0.715192	−0.357596	−	0.933876i	$-0.616404\pi$
$479$	−15.6527	−0.715192	−0.357596	+	0.933876i	$0.616404\pi$
$480$	0	0
$481$	4.16535	0.189924
$482$	0	0
$483$	−0.182026	−0.00828245
$484$	0	0
$485$	30.1033	1.36692
$486$	0	0
$487$	−34.0089	−1.54109	−0.770546	−	0.637385i	$-0.780016\pi$
$487$	−34.0089	−1.54109	−0.770546	+	0.637385i	$0.780016\pi$
$488$	0	0
$489$	−3.51370	−0.158895
$490$	0	0
$491$	13.9859	0.631175	0.315588	−	0.948896i	$-0.397798\pi$
$491$	13.9859	0.631175	0.315588	+	0.948896i	$0.397798\pi$
$492$	0	0
$493$	2.50759	0.112936
$494$	0	0
$495$	−33.9483	−1.52586
$496$	0	0
$497$	−0.251853	−0.0112972
$498$	0	0
$499$	23.0930	1.03379	0.516893	−	0.856050i	$-0.327088\pi$
$499$	23.0930	1.03379	0.516893	+	0.856050i	$0.327088\pi$
$500$	0	0
$501$	14.4054	0.643584
$502$	0	0
$503$	−5.22159	−0.232819	−0.116410	−	0.993201i	$-0.537139\pi$
$503$	−5.22159	−0.232819	−0.116410	+	0.993201i	$0.537139\pi$
$504$	0	0
$505$	18.5066	0.823533
$506$	0	0
$507$	5.60522	0.248937
$508$	0	0
$509$	22.9495	1.01722	0.508609	−	0.860998i	$-0.330160\pi$
$509$	22.9495	1.01722	0.508609	+	0.860998i	$0.330160\pi$
$510$	0	0
$511$	−0.783233	−0.0346482
$512$	0	0
$513$	13.5320	0.597454
$514$	0	0
$515$	8.84891	0.389930
$516$	0	0
$517$	50.5705	2.22409
$518$	0	0
$519$	0.909165	0.0399079
$520$	0	0
$521$	13.5137	0.592046	0.296023	−	0.955181i	$-0.404339\pi$
$521$	13.5137	0.592046	0.296023	+	0.955181i	$0.404339\pi$
$522$	0	0
$523$	7.79860	0.341009	0.170504	−	0.985357i	$-0.445460\pi$
$523$	7.79860	0.341009	0.170504	+	0.985357i	$0.445460\pi$
$524$	0	0
$525$	−0.0440403	−0.00192208
$526$	0	0
$527$	9.58649	0.417594
$528$	0	0
$529$	−10.0472	−0.436834
$530$	0	0
$531$	3.47518	0.150810
$532$	0	0
$533$	10.9911	0.476078
$534$	0	0
$535$	−24.4047	−1.05511
$536$	0	0
$537$	−1.68010	−0.0725017
$538$	0	0
$539$	46.1487	1.98777
$540$	0	0
$541$	8.47910	0.364545	0.182272	−	0.983248i	$-0.441655\pi$
$541$	8.47910	0.364545	0.182272	+	0.983248i	$0.441655\pi$
$542$	0	0
$543$	1.00001	0.0429146
$544$	0	0
$545$	−13.1860	−0.564825
$546$	0	0
$547$	11.0793	0.473717	0.236858	−	0.971544i	$-0.423882\pi$
$547$	11.0793	0.473717	0.236858	+	0.971544i	$0.423882\pi$
$548$	0	0
$549$	−2.49964	−0.106682
$550$	0	0
$551$	−3.17632	−0.135316
$552$	0	0
$553$	1.03286	0.0439217
$554$	0	0
$555$	−2.64112	−0.112109
$556$	0	0
$557$	28.9480	1.22657	0.613283	−	0.789863i	$-0.289848\pi$
$557$	28.9480	1.22657	0.613283	+	0.789863i	$0.289848\pi$
$558$	0	0
$559$	19.1279	0.809025
$560$	0	0
$561$	−12.7410	−0.537928
$562$	0	0
$563$	−31.2581	−1.31737	−0.658685	−	0.752418i	$-0.728887\pi$
$563$	−31.2581	−1.31737	−0.658685	+	0.752418i	$0.728887\pi$
$564$	0	0
$565$	−6.41270	−0.269784
$566$	0	0
$567$	−0.370283	−0.0155504
$568$	0	0
$569$	2.75797	0.115620	0.0578100	−	0.998328i	$-0.481588\pi$
$569$	2.75797	0.115620	0.0578100	+	0.998328i	$0.481588\pi$
$570$	0	0
$571$	34.2703	1.43417	0.717084	−	0.696986i	$-0.245476\pi$
$571$	34.2703	1.43417	0.717084	+	0.696986i	$0.245476\pi$
$572$	0	0
$573$	11.6103	0.485025
$574$	0	0
$575$	3.13388	0.130692
$576$	0	0
$577$	−46.3408	−1.92919	−0.964597	−	0.263728i	$-0.915048\pi$
$577$	−46.3408	−1.92919	−0.964597	+	0.263728i	$0.915048\pi$
$578$	0	0
$579$	−8.92749	−0.371014
$580$	0	0
$581$	0.00822506	0.000341233 0
$582$	0	0
$583$	46.4752	1.92480
$584$	0	0
$585$	−11.2795	−0.466348
$586$	0	0
$587$	−10.5190	−0.434166	−0.217083	−	0.976153i	$-0.569654\pi$
$587$	−10.5190	−0.434166	−0.217083	+	0.976153i	$0.569654\pi$
$588$	0	0
$589$	−12.1430	−0.500345
$590$	0	0
$591$	7.38151	0.303635
$592$	0	0
$593$	−15.6067	−0.640890	−0.320445	−	0.947267i	$-0.603832\pi$
$593$	−15.6067	−0.640890	−0.320445	+	0.947267i	$0.603832\pi$
$594$	0	0
$595$	−0.424174	−0.0173895
$596$	0	0
$597$	9.02560	0.369393
$598$	0	0
$599$	−13.7620	−0.562300	−0.281150	−	0.959664i	$-0.590716\pi$
$599$	−13.7620	−0.562300	−0.281150	+	0.959664i	$0.590716\pi$
$600$	0	0
$601$	31.2077	1.27299	0.636493	−	0.771282i	$-0.280384\pi$
$601$	31.2077	1.27299	0.636493	+	0.771282i	$0.280384\pi$
$602$	0	0
$603$	0.341335	0.0139002
$604$	0	0
$605$	−66.1053	−2.68756
$606$	0	0
$607$	3.52856	0.143220	0.0716100	−	0.997433i	$-0.477186\pi$
$607$	3.52856	0.143220	0.0716100	+	0.997433i	$0.477186\pi$
$608$	0	0
$609$	−0.0449235	−0.00182039
$610$	0	0
$611$	16.8023	0.679746
$612$	0	0
$613$	0.995724	0.0402169	0.0201085	−	0.999798i	$-0.493599\pi$
$613$	0.995724	0.0402169	0.0201085	+	0.999798i	$0.493599\pi$
$614$	0	0
$615$	−6.96912	−0.281022
$616$	0	0
$617$	−10.7787	−0.433936	−0.216968	−	0.976179i	$-0.569617\pi$
$617$	−10.7787	−0.433936	−0.216968	+	0.976179i	$0.569617\pi$
$618$	0	0
$619$	−41.8494	−1.68207	−0.841034	−	0.540982i	$-0.818053\pi$
$619$	−41.8494	−1.68207	−0.841034	+	0.540982i	$0.818053\pi$
$620$	0	0
$621$	−13.6190	−0.546512
$622$	0	0
$623$	0.00849898	0.000340505 0
$624$	0	0
$625$	−19.8880	−0.795518
$626$	0	0
$627$	16.1389	0.644524
$628$	0	0
$629$	5.36430	0.213888
$630$	0	0
$631$	−24.5218	−0.976196	−0.488098	−	0.872789i	$-0.662309\pi$
$631$	−24.5218	−0.976196	−0.488098	+	0.872789i	$0.662309\pi$
$632$	0	0
$633$	3.69415	0.146829
$634$	0	0
$635$	−41.8537	−1.66091
$636$	0	0
$637$	15.3331	0.607519
$638$	0	0
$639$	−8.62486	−0.341194
$640$	0	0
$641$	−23.0254	−0.909447	−0.454724	−	0.890633i	$-0.650262\pi$
$641$	−23.0254	−0.909447	−0.454724	+	0.890633i	$0.650262\pi$
$642$	0	0
$643$	−11.5967	−0.457327	−0.228664	−	0.973505i	$-0.573436\pi$
$643$	−11.5967	−0.457327	−0.228664	+	0.973505i	$0.573436\pi$
$644$	0	0
$645$	−12.1284	−0.477556
$646$	0	0
$647$	5.54198	0.217878	0.108939	−	0.994048i	$-0.465255\pi$
$647$	5.54198	0.217878	0.108939	+	0.994048i	$0.465255\pi$
$648$	0	0
$649$	9.05516	0.355446
$650$	0	0
$651$	−0.171742	−0.00673111
$652$	0	0
$653$	−3.89284	−0.152339	−0.0761693	−	0.997095i	$-0.524269\pi$
$653$	−3.89284	−0.152339	−0.0761693	+	0.997095i	$0.524269\pi$
$654$	0	0
$655$	19.5398	0.763482
$656$	0	0
$657$	−26.8223	−1.04644
$658$	0	0
$659$	−17.9274	−0.698353	−0.349177	−	0.937057i	$-0.613539\pi$
$659$	−17.9274	−0.698353	−0.349177	+	0.937057i	$0.613539\pi$
$660$	0	0
$661$	5.52221	0.214789	0.107395	−	0.994216i	$-0.465749\pi$
$661$	5.52221	0.214789	0.107395	+	0.994216i	$0.465749\pi$
$662$	0	0
$663$	−4.23327	−0.164406
$664$	0	0
$665$	0.537294	0.0208354
$666$	0	0
$667$	3.19673	0.123778
$668$	0	0
$669$	8.07233	0.312094
$670$	0	0
$671$	−6.51323	−0.251440
$672$	0	0
$673$	32.1520	1.23937	0.619684	−	0.784852i	$-0.287261\pi$
$673$	32.1520	1.23937	0.619684	+	0.784852i	$0.287261\pi$
$674$	0	0
$675$	−3.29506	−0.126827
$676$	0	0
$677$	38.6060	1.48375	0.741874	−	0.670539i	$-0.233937\pi$
$677$	38.6060	1.48375	0.741874	+	0.670539i	$0.233937\pi$
$678$	0	0
$679$	1.09536	0.0420360
$680$	0	0
$681$	3.24260	0.124257
$682$	0	0
$683$	−20.8357	−0.797257	−0.398628	−	0.917113i	$-0.630514\pi$
$683$	−20.8357	−0.797257	−0.398628	+	0.917113i	$0.630514\pi$
$684$	0	0
$685$	28.4026	1.08521
$686$	0	0
$687$	5.97119	0.227815
$688$	0	0
$689$	15.4416	0.588277
$690$	0	0
$691$	−39.6858	−1.50972	−0.754860	−	0.655886i	$-0.772295\pi$
$691$	−39.6858	−1.50972	−0.754860	+	0.655886i	$0.772295\pi$
$692$	0	0
$693$	−1.23527	−0.0469239
$694$	0	0
$695$	21.2929	0.807687
$696$	0	0
$697$	14.1548	0.536150
$698$	0	0
$699$	−8.32141	−0.314745
$700$	0	0
$701$	−21.7469	−0.821368	−0.410684	−	0.911778i	$-0.634710\pi$
$701$	−21.7469	−0.821368	−0.410684	+	0.911778i	$0.634710\pi$
$702$	0	0
$703$	−6.79486	−0.256273
$704$	0	0
$705$	−10.6538	−0.401245
$706$	0	0
$707$	0.673394	0.0253256
$708$	0	0
$709$	−10.9993	−0.413086	−0.206543	−	0.978437i	$-0.566221\pi$
$709$	−10.9993	−0.413086	−0.206543	+	0.978437i	$0.566221\pi$
$710$	0	0
$711$	35.3710	1.32651
$712$	0	0
$713$	12.2211	0.457683
$714$	0	0
$715$	−29.3905	−1.09914
$716$	0	0
$717$	13.0469	0.487244
$718$	0	0
$719$	−9.63747	−0.359417	−0.179709	−	0.983720i	$-0.557515\pi$
$719$	−9.63747	−0.359417	−0.179709	+	0.983720i	$0.557515\pi$
$720$	0	0
$721$	0.321983	0.0119913
$722$	0	0
$723$	−15.1733	−0.564300
$724$	0	0
$725$	0.773436	0.0287247
$726$	0	0
$727$	3.59365	0.133281	0.0666406	−	0.997777i	$-0.478772\pi$
$727$	3.59365	0.133281	0.0666406	+	0.997777i	$0.478772\pi$
$728$	0	0
$729$	−4.91527	−0.182047
$730$	0	0
$731$	24.6337	0.911109
$732$	0	0
$733$	6.58136	0.243088	0.121544	−	0.992586i	$-0.461215\pi$
$733$	6.58136	0.243088	0.121544	+	0.992586i	$0.461215\pi$
$734$	0	0
$735$	−9.72223	−0.358610
$736$	0	0
$737$	0.889403	0.0327616
$738$	0	0
$739$	−43.6462	−1.60555	−0.802775	−	0.596282i	$-0.796644\pi$
$739$	−43.6462	−1.60555	−0.802775	+	0.596282i	$0.796644\pi$
$740$	0	0
$741$	5.36220	0.196986
$742$	0	0
$743$	9.15433	0.335840	0.167920	−	0.985801i	$-0.446295\pi$
$743$	9.15433	0.335840	0.167920	+	0.985801i	$0.446295\pi$
$744$	0	0
$745$	−20.9147	−0.766255
$746$	0	0
$747$	0.281672	0.0103058
$748$	0	0
$749$	−0.888008	−0.0324471
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−6.17488	−0.225025
$754$	0	0
$755$	−22.7316	−0.827289
$756$	0	0
$757$	−44.0111	−1.59961	−0.799806	−	0.600259i	$-0.795064\pi$
$757$	−44.0111	−1.59961	−0.799806	+	0.600259i	$0.795064\pi$
$758$	0	0
$759$	−16.2426	−0.589569
$760$	0	0
$761$	23.6173	0.856125	0.428062	−	0.903749i	$-0.359196\pi$
$761$	23.6173	0.856125	0.428062	+	0.903749i	$0.359196\pi$
$762$	0	0
$763$	−0.479794	−0.0173697
$764$	0	0
$765$	−14.5261	−0.525192
$766$	0	0
$767$	3.00862	0.108635
$768$	0	0
$769$	−25.8416	−0.931873	−0.465937	−	0.884818i	$-0.654283\pi$
$769$	−25.8416	−0.931873	−0.465937	+	0.884818i	$0.654283\pi$
$770$	0	0
$771$	−14.7376	−0.530762
$772$	0	0
$773$	44.6722	1.60675	0.803374	−	0.595474i	$-0.203036\pi$
$773$	44.6722	1.60675	0.803374	+	0.595474i	$0.203036\pi$
$774$	0	0
$775$	2.95684	0.106213
$776$	0	0
$777$	−0.0961016	−0.00344763
$778$	0	0
$779$	−17.9296	−0.642395
$780$	0	0
$781$	−22.4735	−0.804165
$782$	0	0
$783$	−3.36114	−0.120118
$784$	0	0
$785$	−49.2159	−1.75659
$786$	0	0
$787$	−10.8667	−0.387357	−0.193679	−	0.981065i	$-0.562042\pi$
$787$	−10.8667	−0.387357	−0.193679	+	0.981065i	$0.562042\pi$
$788$	0	0
$789$	3.04717	0.108482
$790$	0	0
$791$	−0.233337	−0.00829651
$792$	0	0
$793$	−2.16405	−0.0768476
$794$	0	0
$795$	−9.79101	−0.347251
$796$	0	0
$797$	30.0713	1.06518	0.532590	−	0.846373i	$-0.321219\pi$
$797$	30.0713	1.06518	0.532590	+	0.846373i	$0.321219\pi$
$798$	0	0
$799$	21.6386	0.765517
$800$	0	0
$801$	0.291053	0.0102838
$802$	0	0
$803$	−69.8899	−2.46636
$804$	0	0
$805$	−0.540748	−0.0190589
$806$	0	0
$807$	−6.06067	−0.213346
$808$	0	0
$809$	−18.8383	−0.662321	−0.331160	−	0.943574i	$-0.607440\pi$
$809$	−18.8383	−0.662321	−0.331160	+	0.943574i	$0.607440\pi$
$810$	0	0
$811$	−4.65017	−0.163290	−0.0816448	−	0.996661i	$-0.526017\pi$
$811$	−4.65017	−0.163290	−0.0816448	+	0.996661i	$0.526017\pi$
$812$	0	0
$813$	−10.4154	−0.365284
$814$	0	0
$815$	−10.4382	−0.365636
$816$	0	0
$817$	−31.2030	−1.09166
$818$	0	0
$819$	−0.410422	−0.0143413
$820$	0	0
$821$	−20.4667	−0.714292	−0.357146	−	0.934049i	$-0.616250\pi$
$821$	−20.4667	−0.714292	−0.357146	+	0.934049i	$0.616250\pi$
$822$	0	0
$823$	−44.7395	−1.55952	−0.779760	−	0.626079i	$-0.784659\pi$
$823$	−44.7395	−1.55952	−0.779760	+	0.626079i	$0.784659\pi$
$824$	0	0
$825$	−3.92983	−0.136819
$826$	0	0
$827$	−32.4135	−1.12713	−0.563564	−	0.826072i	$-0.690570\pi$
$827$	−32.4135	−1.12713	−0.563564	+	0.826072i	$0.690570\pi$
$828$	0	0
$829$	19.9565	0.693119	0.346559	−	0.938028i	$-0.387350\pi$
$829$	19.9565	0.693119	0.346559	+	0.938028i	$0.387350\pi$
$830$	0	0
$831$	5.19049	0.180056
$832$	0	0
$833$	19.7465	0.684177
$834$	0	0
$835$	42.7944	1.48096
$836$	0	0
$837$	−12.8496	−0.444148
$838$	0	0
$839$	−17.1453	−0.591922	−0.295961	−	0.955200i	$-0.595640\pi$
$839$	−17.1453	−0.591922	−0.295961	+	0.955200i	$0.595640\pi$
$840$	0	0
$841$	−28.2111	−0.972795
$842$	0	0
$843$	18.9395	0.652310
$844$	0	0
$845$	16.6516	0.572831
$846$	0	0
$847$	−2.40535	−0.0826489
$848$	0	0
$849$	1.50040	0.0514935
$850$	0	0
$851$	6.83853	0.234422
$852$	0	0
$853$	−1.20277	−0.0411821	−0.0205911	−	0.999788i	$-0.506555\pi$
$853$	−1.20277	−0.0411821	−0.0205911	+	0.999788i	$0.506555\pi$
$854$	0	0
$855$	18.4000	0.629266
$856$	0	0
$857$	−12.9389	−0.441986	−0.220993	−	0.975275i	$-0.570930\pi$
$857$	−12.9389	−0.441986	−0.220993	+	0.975275i	$0.570930\pi$
$858$	0	0
$859$	−17.9312	−0.611805	−0.305903	−	0.952063i	$-0.598958\pi$
$859$	−17.9312	−0.611805	−0.305903	+	0.952063i	$0.598958\pi$
$860$	0	0
$861$	−0.253583	−0.00864210
$862$	0	0
$863$	41.2906	1.40555	0.702774	−	0.711413i	$-0.251945\pi$
$863$	41.2906	1.40555	0.702774	+	0.711413i	$0.251945\pi$
$864$	0	0
$865$	2.70088	0.0918326
$866$	0	0
$867$	6.17668	0.209771
$868$	0	0
$869$	92.1648	3.12648
$870$	0	0
$871$	0.295508	0.0100129
$872$	0	0
$873$	37.5112	1.26956
$874$	0	0
$875$	−0.882078	−0.0298197
$876$	0	0
$877$	−34.1316	−1.15254	−0.576271	−	0.817259i	$-0.695493\pi$
$877$	−34.1316	−1.15254	−0.576271	+	0.817259i	$0.695493\pi$
$878$	0	0
$879$	16.6601	0.561931
$880$	0	0
$881$	−33.9088	−1.14242	−0.571208	−	0.820805i	$-0.693525\pi$
$881$	−33.9088	−1.14242	−0.571208	+	0.820805i	$0.693525\pi$
$882$	0	0
$883$	−4.13156	−0.139038	−0.0695190	−	0.997581i	$-0.522146\pi$
$883$	−4.13156	−0.139038	−0.0695190	+	0.997581i	$0.522146\pi$
$884$	0	0
$885$	−1.90767	−0.0641256
$886$	0	0
$887$	52.9804	1.77891	0.889453	−	0.457026i	$-0.151086\pi$
$887$	52.9804	1.77891	0.889453	+	0.457026i	$0.151086\pi$
$888$	0	0
$889$	−1.52292	−0.0510771
$890$	0	0
$891$	−33.0412	−1.10692
$892$	0	0
$893$	−27.4092	−0.917214
$894$	0	0
$895$	−4.99112	−0.166835
$896$	0	0
$897$	−5.39667	−0.180190
$898$	0	0
$899$	3.01614	0.100594
$900$	0	0
$901$	19.8862	0.662506
$902$	0	0
$903$	−0.441313	−0.0146860
$904$	0	0
$905$	2.97076	0.0987513
$906$	0	0
$907$	29.5783	0.982130	0.491065	−	0.871123i	$-0.336608\pi$
$907$	29.5783	0.982130	0.491065	+	0.871123i	$0.336608\pi$
$908$	0	0
$909$	23.0608	0.764878
$910$	0	0
$911$	39.3813	1.30476	0.652381	−	0.757891i	$-0.273770\pi$
$911$	39.3813	1.30476	0.652381	+	0.757891i	$0.273770\pi$
$912$	0	0
$913$	0.733943	0.0242900
$914$	0	0
$915$	1.37215	0.0453620
$916$	0	0
$917$	0.710987	0.0234789
$918$	0	0
$919$	4.83998	0.159656	0.0798282	−	0.996809i	$-0.474563\pi$
$919$	4.83998	0.159656	0.0798282	+	0.996809i	$0.474563\pi$
$920$	0	0
$921$	7.32454	0.241352
$922$	0	0
$923$	−7.46691	−0.245776
$924$	0	0
$925$	1.65455	0.0544014
$926$	0	0
$927$	11.0265	0.362158
$928$	0	0
$929$	−30.7096	−1.00755	−0.503775	−	0.863835i	$-0.668056\pi$
$929$	−30.7096	−1.00755	−0.503775	+	0.863835i	$0.668056\pi$
$930$	0	0
$931$	−25.0126	−0.819754
$932$	0	0
$933$	−9.86545	−0.322980
$934$	0	0
$935$	−37.8501	−1.23783
$936$	0	0
$937$	−48.5000	−1.58443	−0.792213	−	0.610245i	$-0.791071\pi$
$937$	−48.5000	−1.58443	−0.792213	+	0.610245i	$0.791071\pi$
$938$	0	0
$939$	−3.33091	−0.108700
$940$	0	0
$941$	−42.2803	−1.37830	−0.689150	−	0.724619i	$-0.742016\pi$
$941$	−42.2803	−1.37830	−0.689150	+	0.724619i	$0.742016\pi$
$942$	0	0
$943$	18.0448	0.587621
$944$	0	0
$945$	0.568559	0.0184952
$946$	0	0
$947$	7.93645	0.257900	0.128950	−	0.991651i	$-0.458839\pi$
$947$	7.93645	0.257900	0.128950	+	0.991651i	$0.458839\pi$
$948$	0	0
$949$	−23.2212	−0.753792
$950$	0	0
$951$	−9.51198	−0.308447
$952$	0	0
$953$	−50.2558	−1.62795	−0.813973	−	0.580903i	$-0.802700\pi$
$953$	−50.2558	−1.62795	−0.813973	+	0.580903i	$0.802700\pi$
$954$	0	0
$955$	34.4909	1.11610
$956$	0	0
$957$	−4.00864	−0.129581
$958$	0	0
$959$	1.03348	0.0333727
$960$	0	0
$961$	−19.4693	−0.628043
$962$	0	0
$963$	−30.4104	−0.979960
$964$	0	0
$965$	−26.5211	−0.853745
$966$	0	0
$967$	45.7638	1.47166	0.735832	−	0.677164i	$-0.236791\pi$
$967$	45.7638	1.47166	0.735832	+	0.677164i	$0.236791\pi$
$968$	0	0
$969$	6.90565	0.221841
$970$	0	0
$971$	−59.5742	−1.91183	−0.955914	−	0.293647i	$-0.905131\pi$
$971$	−59.5742	−1.91183	−0.955914	+	0.293647i	$0.905131\pi$
$972$	0	0
$973$	0.774779	0.0248383
$974$	0	0
$975$	−1.30570	−0.0418159
$976$	0	0
$977$	−35.2079	−1.12640	−0.563200	−	0.826321i	$-0.690430\pi$
$977$	−35.2079	−1.12640	−0.563200	+	0.826321i	$0.690430\pi$
$978$	0	0
$979$	0.758386	0.0242381
$980$	0	0
$981$	−16.4308	−0.524596
$982$	0	0
$983$	21.7034	0.692231	0.346115	−	0.938192i	$-0.387501\pi$
$983$	21.7034	0.692231	0.346115	+	0.938192i	$0.387501\pi$
$984$	0	0
$985$	21.9284	0.698698
$986$	0	0
$987$	−0.387656	−0.0123392
$988$	0	0
$989$	31.4036	0.998576
$990$	0	0
$991$	−40.2004	−1.27701	−0.638504	−	0.769619i	$-0.720446\pi$
$991$	−40.2004	−1.27701	−0.638504	+	0.769619i	$0.720446\pi$
$992$	0	0
$993$	−2.48498	−0.0788585
$994$	0	0
$995$	26.8126	0.850016
$996$	0	0
$997$	−32.0912	−1.01634	−0.508169	−	0.861257i	$-0.669678\pi$
$997$	−32.0912	−1.01634	−0.508169	+	0.861257i	$0.669678\pi$
$998$	0	0
$999$	−7.19025	−0.227489

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.20	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.20	✓	50	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.684026 q^{3} -2.03205 q^{5} -0.0739397 q^{7} -2.53211 q^{9} +O(q^{10})\)	\(q-0.684026 q^{3} -2.03205 q^{5} -0.0739397 q^{7} -2.53211 q^{9} -6.59782 q^{11} -2.19215 q^{13} +1.38998 q^{15} -2.82314 q^{17} +3.57602 q^{19} +0.0505766 q^{21} -3.59900 q^{23} -0.870764 q^{25} +3.78411 q^{27} -0.888227 q^{29} -3.39569 q^{31} +4.51308 q^{33} +0.150249 q^{35} -1.90012 q^{37} +1.49949 q^{39} -5.01384 q^{41} -8.72563 q^{43} +5.14538 q^{45} -7.66472 q^{47} -6.99453 q^{49} +1.93110 q^{51} -7.04401 q^{53} +13.4071 q^{55} -2.44609 q^{57} -1.37245 q^{59} +0.987178 q^{61} +0.187223 q^{63} +4.45457 q^{65} -0.134803 q^{67} +2.46181 q^{69} +3.40620 q^{71} +10.5929 q^{73} +0.595625 q^{75} +0.487841 q^{77} -13.9690 q^{79} +5.00790 q^{81} -0.111240 q^{83} +5.73676 q^{85} +0.607570 q^{87} -0.114945 q^{89} +0.162087 q^{91} +2.32274 q^{93} -7.26666 q^{95} -14.8142 q^{97} +16.7064 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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