LMFDB - Embedded newform 6008.2.a.e.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.83142 q^{3} +4.26484 q^{5} +3.89972 q^{7} +5.01693 q^{9} +O(q^{10})$	$q-2.83142 q^{3} +4.26484 q^{5} +3.89972 q^{7} +5.01693 q^{9} +2.11918 q^{11} +2.93611 q^{13} -12.0755 q^{15} -2.84435 q^{17} -7.25277 q^{19} -11.0417 q^{21} +0.341822 q^{23} +13.1888 q^{25} -5.71078 q^{27} +0.780311 q^{29} +5.53879 q^{31} -6.00029 q^{33} +16.6317 q^{35} +6.82954 q^{37} -8.31336 q^{39} -8.86143 q^{41} -4.55369 q^{43} +21.3964 q^{45} -9.00906 q^{47} +8.20784 q^{49} +8.05354 q^{51} +4.97585 q^{53} +9.03796 q^{55} +20.5356 q^{57} +1.52810 q^{59} +7.97529 q^{61} +19.5646 q^{63} +12.5220 q^{65} +10.5064 q^{67} -0.967841 q^{69} +13.1735 q^{71} +8.40366 q^{73} -37.3432 q^{75} +8.26421 q^{77} -2.38763 q^{79} +1.11882 q^{81} -10.0730 q^{83} -12.1307 q^{85} -2.20939 q^{87} +3.91403 q^{89} +11.4500 q^{91} -15.6826 q^{93} -30.9319 q^{95} +4.05224 q^{97} +10.6318 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.83142	−1.63472	−0.817360	−	0.576127i	$-0.804563\pi$
$3$	−2.83142	−1.63472	−0.817360	+	0.576127i	$0.804563\pi$
$4$	0	0
$5$	4.26484	1.90729	0.953647	−	0.300928i	$-0.0972964\pi$
$5$	4.26484	1.90729	0.953647	+	0.300928i	$0.0972964\pi$
$6$	0	0
$7$	3.89972	1.47396	0.736978	−	0.675916i	$-0.236252\pi$
$7$	3.89972	1.47396	0.736978	+	0.675916i	$0.236252\pi$
$8$	0	0
$9$	5.01693	1.67231
$10$	0	0
$11$	2.11918	0.638957	0.319478	−	0.947594i	$-0.396492\pi$
$11$	2.11918	0.638957	0.319478	+	0.947594i	$0.396492\pi$
$12$	0	0
$13$	2.93611	0.814330	0.407165	−	0.913355i	$-0.366517\pi$
$13$	2.93611	0.814330	0.407165	+	0.913355i	$0.366517\pi$
$14$	0	0
$15$	−12.0755	−3.11789
$16$	0	0
$17$	−2.84435	−0.689856	−0.344928	−	0.938629i	$-0.612097\pi$
$17$	−2.84435	−0.689856	−0.344928	+	0.938629i	$0.612097\pi$
$18$	0	0
$19$	−7.25277	−1.66390	−0.831950	−	0.554850i	$-0.812776\pi$
$19$	−7.25277	−1.66390	−0.831950	+	0.554850i	$0.812776\pi$
$20$	0	0
$21$	−11.0417	−2.40951
$22$	0	0
$23$	0.341822	0.0712748	0.0356374	−	0.999365i	$-0.488654\pi$
$23$	0.341822	0.0712748	0.0356374	+	0.999365i	$0.488654\pi$
$24$	0	0
$25$	13.1888	2.63777
$26$	0	0
$27$	−5.71078	−1.09904
$28$	0	0
$29$	0.780311	0.144900	0.0724501	−	0.997372i	$-0.476918\pi$
$29$	0.780311	0.144900	0.0724501	+	0.997372i	$0.476918\pi$
$30$	0	0
$31$	5.53879	0.994796	0.497398	−	0.867522i	$-0.334289\pi$
$31$	5.53879	0.994796	0.497398	+	0.867522i	$0.334289\pi$
$32$	0	0
$33$	−6.00029	−1.04452
$34$	0	0
$35$	16.6317	2.81127
$36$	0	0
$37$	6.82954	1.12277	0.561385	−	0.827555i	$-0.310269\pi$
$37$	6.82954	1.12277	0.561385	+	0.827555i	$0.310269\pi$
$38$	0	0
$39$	−8.31336	−1.33120
$40$	0	0
$41$	−8.86143	−1.38392	−0.691962	−	0.721934i	$-0.743253\pi$
$41$	−8.86143	−1.38392	−0.691962	+	0.721934i	$0.743253\pi$
$42$	0	0
$43$	−4.55369	−0.694431	−0.347215	−	0.937785i	$-0.612873\pi$
$43$	−4.55369	−0.694431	−0.347215	+	0.937785i	$0.612873\pi$
$44$	0	0
$45$	21.3964	3.18959
$46$	0	0
$47$	−9.00906	−1.31411	−0.657053	−	0.753844i	$-0.728197\pi$
$47$	−9.00906	−1.31411	−0.657053	+	0.753844i	$0.728197\pi$
$48$	0	0
$49$	8.20784	1.17255
$50$	0	0
$51$	8.05354	1.12772
$52$	0	0
$53$	4.97585	0.683485	0.341742	−	0.939794i	$-0.388983\pi$
$53$	4.97585	0.683485	0.341742	+	0.939794i	$0.388983\pi$
$54$	0	0
$55$	9.03796	1.21868
$56$	0	0
$57$	20.5356	2.72001
$58$	0	0
$59$	1.52810	0.198941	0.0994707	−	0.995040i	$-0.468285\pi$
$59$	1.52810	0.198941	0.0994707	+	0.995040i	$0.468285\pi$
$60$	0	0
$61$	7.97529	1.02113	0.510566	−	0.859839i	$-0.329436\pi$
$61$	7.97529	1.02113	0.510566	+	0.859839i	$0.329436\pi$
$62$	0	0
$63$	19.5646	2.46491
$64$	0	0
$65$	12.5220	1.55317
$66$	0	0
$67$	10.5064	1.28356	0.641778	−	0.766891i	$-0.278197\pi$
$67$	10.5064	1.28356	0.641778	+	0.766891i	$0.278197\pi$
$68$	0	0
$69$	−0.967841	−0.116514
$70$	0	0
$71$	13.1735	1.56341	0.781706	−	0.623648i	$-0.214350\pi$
$71$	13.1735	1.56341	0.781706	+	0.623648i	$0.214350\pi$
$72$	0	0
$73$	8.40366	0.983574	0.491787	−	0.870716i	$-0.336344\pi$
$73$	8.40366	0.983574	0.491787	+	0.870716i	$0.336344\pi$
$74$	0	0
$75$	−37.3432	−4.31202
$76$	0	0
$77$	8.26421	0.941795
$78$	0	0
$79$	−2.38763	−0.268630	−0.134315	−	0.990939i	$-0.542883\pi$
$79$	−2.38763	−0.268630	−0.134315	+	0.990939i	$0.542883\pi$
$80$	0	0
$81$	1.11882	0.124313
$82$	0	0
$83$	−10.0730	−1.10566	−0.552830	−	0.833294i	$-0.686452\pi$
$83$	−10.0730	−1.10566	−0.552830	+	0.833294i	$0.686452\pi$
$84$	0	0
$85$	−12.1307	−1.31576
$86$	0	0
$87$	−2.20939	−0.236871
$88$	0	0
$89$	3.91403	0.414886	0.207443	−	0.978247i	$-0.433486\pi$
$89$	3.91403	0.414886	0.207443	+	0.978247i	$0.433486\pi$
$90$	0	0
$91$	11.4500	1.20029
$92$	0	0
$93$	−15.6826	−1.62621
$94$	0	0
$95$	−30.9319	−3.17355
$96$	0	0
$97$	4.05224	0.411442	0.205721	−	0.978611i	$-0.434046\pi$
$97$	4.05224	0.411442	0.205721	+	0.978611i	$0.434046\pi$
$98$	0	0
$99$	10.6318	1.06853
$100$	0	0
$101$	12.3529	1.22916	0.614580	−	0.788854i	$-0.289325\pi$
$101$	12.3529	1.22916	0.614580	+	0.788854i	$0.289325\pi$
$102$	0	0
$103$	18.1312	1.78652	0.893262	−	0.449536i	$-0.148411\pi$
$103$	18.1312	1.78652	0.893262	+	0.449536i	$0.148411\pi$
$104$	0	0
$105$	−47.0913	−4.59564
$106$	0	0
$107$	−17.8437	−1.72501	−0.862506	−	0.506047i	$-0.831106\pi$
$107$	−17.8437	−1.72501	−0.862506	+	0.506047i	$0.831106\pi$
$108$	0	0
$109$	17.0803	1.63600	0.817999	−	0.575220i	$-0.195084\pi$
$109$	17.0803	1.63600	0.817999	+	0.575220i	$0.195084\pi$
$110$	0	0
$111$	−19.3373	−1.83541
$112$	0	0
$113$	−9.34564	−0.879164	−0.439582	−	0.898203i	$-0.644873\pi$
$113$	−9.34564	−0.879164	−0.439582	+	0.898203i	$0.644873\pi$
$114$	0	0
$115$	1.45782	0.135942
$116$	0	0
$117$	14.7303	1.36181
$118$	0	0
$119$	−11.0922	−1.01682
$120$	0	0
$121$	−6.50908	−0.591734
$122$	0	0
$123$	25.0904	2.26233
$124$	0	0
$125$	34.9241	3.12371
$126$	0	0
$127$	−16.2726	−1.44396	−0.721982	−	0.691912i	$-0.756769\pi$
$127$	−16.2726	−1.44396	−0.721982	+	0.691912i	$0.756769\pi$
$128$	0	0
$129$	12.8934	1.13520
$130$	0	0
$131$	0.0185339	0.00161931	0.000809655	−	1.00000i	$-0.499742\pi$
$131$	0.0185339	0.00161931	0.000809655		1.00000i	$0.499742\pi$
$132$	0	0
$133$	−28.2838	−2.45252
$134$	0	0
$135$	−24.3556	−2.09619
$136$	0	0
$137$	1.93835	0.165605	0.0828023	−	0.996566i	$-0.473613\pi$
$137$	1.93835	0.165605	0.0828023	+	0.996566i	$0.473613\pi$
$138$	0	0
$139$	−11.6399	−0.987284	−0.493642	−	0.869665i	$-0.664335\pi$
$139$	−11.6399	−0.987284	−0.493642	+	0.869665i	$0.664335\pi$
$140$	0	0
$141$	25.5084	2.14820
$142$	0	0
$143$	6.22215	0.520322
$144$	0	0
$145$	3.32790	0.276367
$146$	0	0
$147$	−23.2398	−1.91679
$148$	0	0
$149$	2.32221	0.190243	0.0951215	−	0.995466i	$-0.469676\pi$
$149$	2.32221	0.190243	0.0951215	+	0.995466i	$0.469676\pi$
$150$	0	0
$151$	−10.9111	−0.887933	−0.443966	−	0.896043i	$-0.646429\pi$
$151$	−10.9111	−0.887933	−0.443966	+	0.896043i	$0.646429\pi$
$152$	0	0
$153$	−14.2699	−1.15365
$154$	0	0
$155$	23.6221	1.89737
$156$	0	0
$157$	−10.3549	−0.826413	−0.413207	−	0.910637i	$-0.635591\pi$
$157$	−10.3549	−0.826413	−0.413207	+	0.910637i	$0.635591\pi$
$158$	0	0
$159$	−14.0887	−1.11731
$160$	0	0
$161$	1.33301	0.105056
$162$	0	0
$163$	−3.98006	−0.311742	−0.155871	−	0.987777i	$-0.549818\pi$
$163$	−3.98006	−0.311742	−0.155871	+	0.987777i	$0.549818\pi$
$164$	0	0
$165$	−25.5903	−1.99220
$166$	0	0
$167$	15.3238	1.18579	0.592894	−	0.805280i	$-0.297985\pi$
$167$	15.3238	1.18579	0.592894	+	0.805280i	$0.297985\pi$
$168$	0	0
$169$	−4.37926	−0.336866
$170$	0	0
$171$	−36.3867	−2.78256
$172$	0	0
$173$	6.23052	0.473697	0.236849	−	0.971547i	$-0.423885\pi$
$173$	6.23052	0.473697	0.236849	+	0.971547i	$0.423885\pi$
$174$	0	0
$175$	51.4328	3.88796
$176$	0	0
$177$	−4.32668	−0.325214
$178$	0	0
$179$	1.69979	0.127049	0.0635243	−	0.997980i	$-0.479766\pi$
$179$	1.69979	0.127049	0.0635243	+	0.997980i	$0.479766\pi$
$180$	0	0
$181$	−13.5760	−1.00910	−0.504549	−	0.863383i	$-0.668341\pi$
$181$	−13.5760	−1.00910	−0.504549	+	0.863383i	$0.668341\pi$
$182$	0	0
$183$	−22.5814	−1.66926
$184$	0	0
$185$	29.1269	2.14145
$186$	0	0
$187$	−6.02769	−0.440788
$188$	0	0
$189$	−22.2705	−1.61994
$190$	0	0
$191$	−14.5539	−1.05308	−0.526542	−	0.850149i	$-0.676512\pi$
$191$	−14.5539	−1.05308	−0.526542	+	0.850149i	$0.676512\pi$
$192$	0	0
$193$	−24.9503	−1.79596	−0.897981	−	0.440034i	$-0.854966\pi$
$193$	−24.9503	−1.79596	−0.897981	+	0.440034i	$0.854966\pi$
$194$	0	0
$195$	−35.4551	−2.53899
$196$	0	0
$197$	−19.0165	−1.35487	−0.677436	−	0.735582i	$-0.736909\pi$
$197$	−19.0165	−1.35487	−0.677436	+	0.735582i	$0.736909\pi$
$198$	0	0
$199$	19.9265	1.41255	0.706277	−	0.707935i	$-0.250373\pi$
$199$	19.9265	1.41255	0.706277	+	0.707935i	$0.250373\pi$
$200$	0	0
$201$	−29.7479	−2.09825
$202$	0	0
$203$	3.04300	0.213576
$204$	0	0
$205$	−37.7926	−2.63955
$206$	0	0
$207$	1.71490	0.119194
$208$	0	0
$209$	−15.3699	−1.06316
$210$	0	0
$211$	23.9950	1.65189	0.825943	−	0.563754i	$-0.190643\pi$
$211$	23.9950	1.65189	0.825943	+	0.563754i	$0.190643\pi$
$212$	0	0
$213$	−37.2998	−2.55574
$214$	0	0
$215$	−19.4207	−1.32448
$216$	0	0
$217$	21.5998	1.46629
$218$	0	0
$219$	−23.7943	−1.60787
$220$	0	0
$221$	−8.35132	−0.561770
$222$	0	0
$223$	8.43959	0.565157	0.282579	−	0.959244i	$-0.408810\pi$
$223$	8.43959	0.565157	0.282579	+	0.959244i	$0.408810\pi$
$224$	0	0
$225$	66.1676	4.41117
$226$	0	0
$227$	5.51956	0.366346	0.183173	−	0.983081i	$-0.441363\pi$
$227$	5.51956	0.366346	0.183173	+	0.983081i	$0.441363\pi$
$228$	0	0
$229$	16.8298	1.11214	0.556072	−	0.831134i	$-0.312308\pi$
$229$	16.8298	1.11214	0.556072	+	0.831134i	$0.312308\pi$
$230$	0	0
$231$	−23.3995	−1.53957
$232$	0	0
$233$	−5.63332	−0.369051	−0.184526	−	0.982828i	$-0.559075\pi$
$233$	−5.63332	−0.369051	−0.184526	+	0.982828i	$0.559075\pi$
$234$	0	0
$235$	−38.4222	−2.50639
$236$	0	0
$237$	6.76039	0.439135
$238$	0	0
$239$	−23.2855	−1.50621	−0.753106	−	0.657900i	$-0.771445\pi$
$239$	−23.2855	−1.50621	−0.753106	+	0.657900i	$0.771445\pi$
$240$	0	0
$241$	−22.5636	−1.45345	−0.726724	−	0.686929i	$-0.758958\pi$
$241$	−22.5636	−1.45345	−0.726724	+	0.686929i	$0.758958\pi$
$242$	0	0
$243$	13.9645	0.895823
$244$	0	0
$245$	35.0051	2.23639
$246$	0	0
$247$	−21.2949	−1.35496
$248$	0	0
$249$	28.5210	1.80744
$250$	0	0
$251$	14.5747	0.919947	0.459974	−	0.887933i	$-0.347859\pi$
$251$	14.5747	0.919947	0.459974	+	0.887933i	$0.347859\pi$
$252$	0	0
$253$	0.724382	0.0455415
$254$	0	0
$255$	34.3471	2.15090
$256$	0	0
$257$	12.5815	0.784810	0.392405	−	0.919793i	$-0.371643\pi$
$257$	12.5815	0.784810	0.392405	+	0.919793i	$0.371643\pi$
$258$	0	0
$259$	26.6333	1.65491
$260$	0	0
$261$	3.91477	0.242318
$262$	0	0
$263$	−4.60784	−0.284132	−0.142066	−	0.989857i	$-0.545374\pi$
$263$	−4.60784	−0.284132	−0.142066	+	0.989857i	$0.545374\pi$
$264$	0	0
$265$	21.2212	1.30361
$266$	0	0
$267$	−11.0823	−0.678223
$268$	0	0
$269$	11.5415	0.703698	0.351849	−	0.936057i	$-0.385553\pi$
$269$	11.5415	0.703698	0.351849	+	0.936057i	$0.385553\pi$
$270$	0	0
$271$	29.6168	1.79909	0.899547	−	0.436823i	$-0.143897\pi$
$271$	29.6168	1.79909	0.899547	+	0.436823i	$0.143897\pi$
$272$	0	0
$273$	−32.4198	−1.96213
$274$	0	0
$275$	27.9495	1.68542
$276$	0	0
$277$	28.2029	1.69455	0.847273	−	0.531158i	$-0.178243\pi$
$277$	28.2029	1.69455	0.847273	+	0.531158i	$0.178243\pi$
$278$	0	0
$279$	27.7877	1.66361
$280$	0	0
$281$	1.55832	0.0929616	0.0464808	−	0.998919i	$-0.485199\pi$
$281$	1.55832	0.0929616	0.0464808	+	0.998919i	$0.485199\pi$
$282$	0	0
$283$	−3.61488	−0.214883	−0.107441	−	0.994211i	$-0.534266\pi$
$283$	−3.61488	−0.214883	−0.107441	+	0.994211i	$0.534266\pi$
$284$	0	0
$285$	87.5812	5.18786
$286$	0	0
$287$	−34.5571	−2.03984
$288$	0	0
$289$	−8.90968	−0.524099
$290$	0	0
$291$	−11.4736	−0.672593
$292$	0	0
$293$	15.6850	0.916327	0.458163	−	0.888868i	$-0.348507\pi$
$293$	15.6850	0.916327	0.458163	+	0.888868i	$0.348507\pi$
$294$	0	0
$295$	6.51709	0.379440
$296$	0	0
$297$	−12.1022	−0.702239
$298$	0	0
$299$	1.00363	0.0580412
$300$	0	0
$301$	−17.7581	−1.02356
$302$	0	0
$303$	−34.9763	−2.00933
$304$	0	0
$305$	34.0133	1.94760
$306$	0	0
$307$	25.4816	1.45431	0.727155	−	0.686473i	$-0.240842\pi$
$307$	25.4816	1.45431	0.727155	+	0.686473i	$0.240842\pi$
$308$	0	0
$309$	−51.3371	−2.92047
$310$	0	0
$311$	26.7911	1.51918	0.759592	−	0.650400i	$-0.225399\pi$
$311$	26.7911	1.51918	0.759592	+	0.650400i	$0.225399\pi$
$312$	0	0
$313$	32.6430	1.84509	0.922546	−	0.385886i	$-0.126104\pi$
$313$	32.6430	1.84509	0.922546	+	0.385886i	$0.126104\pi$
$314$	0	0
$315$	83.4401	4.70131
$316$	0	0
$317$	5.25413	0.295101	0.147551	−	0.989054i	$-0.452861\pi$
$317$	5.25413	0.295101	0.147551	+	0.989054i	$0.452861\pi$
$318$	0	0
$319$	1.65362	0.0925849
$320$	0	0
$321$	50.5229	2.81991
$322$	0	0
$323$	20.6294	1.14785
$324$	0	0
$325$	38.7239	2.14802
$326$	0	0
$327$	−48.3615	−2.67440
$328$	0	0
$329$	−35.1328	−1.93694
$330$	0	0
$331$	−20.8852	−1.14795	−0.573976	−	0.818872i	$-0.694600\pi$
$331$	−20.8852	−1.14795	−0.573976	+	0.818872i	$0.694600\pi$
$332$	0	0
$333$	34.2634	1.87762
$334$	0	0
$335$	44.8079	2.44812
$336$	0	0
$337$	6.38006	0.347544	0.173772	−	0.984786i	$-0.444404\pi$
$337$	6.38006	0.347544	0.173772	+	0.984786i	$0.444404\pi$
$338$	0	0
$339$	26.4614	1.43719
$340$	0	0
$341$	11.7377	0.635632
$342$	0	0
$343$	4.71022	0.254328
$344$	0	0
$345$	−4.12769	−0.222227
$346$	0	0
$347$	−6.57016	−0.352705	−0.176352	−	0.984327i	$-0.556430\pi$
$347$	−6.57016	−0.352705	−0.176352	+	0.984327i	$0.556430\pi$
$348$	0	0
$349$	−15.2031	−0.813801	−0.406901	−	0.913472i	$-0.633390\pi$
$349$	−15.2031	−0.813801	−0.406901	+	0.913472i	$0.633390\pi$
$350$	0	0
$351$	−16.7675	−0.894982
$352$	0	0
$353$	−15.7242	−0.836916	−0.418458	−	0.908236i	$-0.637429\pi$
$353$	−15.7242	−0.836916	−0.418458	+	0.908236i	$0.637429\pi$
$354$	0	0
$355$	56.1830	2.98188
$356$	0	0
$357$	31.4066	1.66221
$358$	0	0
$359$	−25.5096	−1.34635	−0.673174	−	0.739484i	$-0.735069\pi$
$359$	−25.5096	−1.34635	−0.673174	+	0.739484i	$0.735069\pi$
$360$	0	0
$361$	33.6027	1.76857
$362$	0	0
$363$	18.4299	0.967320
$364$	0	0
$365$	35.8403	1.87596
$366$	0	0
$367$	−8.13935	−0.424871	−0.212435	−	0.977175i	$-0.568139\pi$
$367$	−8.13935	−0.424871	−0.212435	+	0.977175i	$0.568139\pi$
$368$	0	0
$369$	−44.4572	−2.31435
$370$	0	0
$371$	19.4044	1.00743
$372$	0	0
$373$	8.29033	0.429257	0.214628	−	0.976696i	$-0.431146\pi$
$373$	8.29033	0.429257	0.214628	+	0.976696i	$0.431146\pi$
$374$	0	0
$375$	−98.8848	−5.10639
$376$	0	0
$377$	2.29108	0.117997
$378$	0	0
$379$	13.5481	0.695921	0.347960	−	0.937509i	$-0.386874\pi$
$379$	13.5481	0.695921	0.347960	+	0.937509i	$0.386874\pi$
$380$	0	0
$381$	46.0747	2.36048
$382$	0	0
$383$	−21.0800	−1.07714	−0.538568	−	0.842582i	$-0.681034\pi$
$383$	−21.0800	−1.07714	−0.538568	+	0.842582i	$0.681034\pi$
$384$	0	0
$385$	35.2455	1.79628
$386$	0	0
$387$	−22.8455	−1.16130
$388$	0	0
$389$	−9.07613	−0.460178	−0.230089	−	0.973170i	$-0.573902\pi$
$389$	−9.07613	−0.460178	−0.230089	+	0.973170i	$0.573902\pi$
$390$	0	0
$391$	−0.972261	−0.0491693
$392$	0	0
$393$	−0.0524771	−0.00264712
$394$	0	0
$395$	−10.1829	−0.512356
$396$	0	0
$397$	−16.3595	−0.821060	−0.410530	−	0.911847i	$-0.634656\pi$
$397$	−16.3595	−0.821060	−0.410530	+	0.911847i	$0.634656\pi$
$398$	0	0
$399$	80.0833	4.00918
$400$	0	0
$401$	21.7143	1.08436	0.542180	−	0.840263i	$-0.317599\pi$
$401$	21.7143	1.08436	0.542180	+	0.840263i	$0.317599\pi$
$402$	0	0
$403$	16.2625	0.810093
$404$	0	0
$405$	4.77157	0.237101
$406$	0	0
$407$	14.4730	0.717402
$408$	0	0
$409$	−33.0374	−1.63360	−0.816798	−	0.576924i	$-0.804253\pi$
$409$	−33.0374	−1.63360	−0.816798	+	0.576924i	$0.804253\pi$
$410$	0	0
$411$	−5.48829	−0.270717
$412$	0	0
$413$	5.95916	0.293231
$414$	0	0
$415$	−42.9599	−2.10882
$416$	0	0
$417$	32.9574	1.61393
$418$	0	0
$419$	−28.2484	−1.38003	−0.690013	−	0.723797i	$-0.742395\pi$
$419$	−28.2484	−1.38003	−0.690013	+	0.723797i	$0.742395\pi$
$420$	0	0
$421$	−7.83979	−0.382088	−0.191044	−	0.981581i	$-0.561187\pi$
$421$	−7.83979	−0.382088	−0.191044	+	0.981581i	$0.561187\pi$
$422$	0	0
$423$	−45.1978	−2.19759
$424$	0	0
$425$	−37.5137	−1.81968
$426$	0	0
$427$	31.1014	1.50510
$428$	0	0
$429$	−17.6175	−0.850581
$430$	0	0
$431$	−13.0936	−0.630697	−0.315348	−	0.948976i	$-0.602121\pi$
$431$	−13.0936	−0.630697	−0.315348	+	0.948976i	$0.602121\pi$
$432$	0	0
$433$	3.22919	0.155185	0.0775926	−	0.996985i	$-0.475277\pi$
$433$	3.22919	0.155185	0.0775926	+	0.996985i	$0.475277\pi$
$434$	0	0
$435$	−9.42268	−0.451783
$436$	0	0
$437$	−2.47916	−0.118594
$438$	0	0
$439$	−10.1627	−0.485039	−0.242520	−	0.970147i	$-0.577974\pi$
$439$	−10.1627	−0.485039	−0.242520	+	0.970147i	$0.577974\pi$
$440$	0	0
$441$	41.1782	1.96086
$442$	0	0
$443$	−1.09206	−0.0518855	−0.0259427	−	0.999663i	$-0.508259\pi$
$443$	−1.09206	−0.0518855	−0.0259427	+	0.999663i	$0.508259\pi$
$444$	0	0
$445$	16.6927	0.791310
$446$	0	0
$447$	−6.57516	−0.310994
$448$	0	0
$449$	9.18522	0.433477	0.216739	−	0.976230i	$-0.430458\pi$
$449$	9.18522	0.433477	0.216739	+	0.976230i	$0.430458\pi$
$450$	0	0
$451$	−18.7790	−0.884267
$452$	0	0
$453$	30.8939	1.45152
$454$	0	0
$455$	48.8325	2.28930
$456$	0	0
$457$	−1.62771	−0.0761412	−0.0380706	−	0.999275i	$-0.512121\pi$
$457$	−1.62771	−0.0761412	−0.0380706	+	0.999275i	$0.512121\pi$
$458$	0	0
$459$	16.2435	0.758179
$460$	0	0
$461$	−16.6795	−0.776842	−0.388421	−	0.921482i	$-0.626979\pi$
$461$	−16.6795	−0.776842	−0.388421	+	0.921482i	$0.626979\pi$
$462$	0	0
$463$	4.78496	0.222376	0.111188	−	0.993799i	$-0.464534\pi$
$463$	4.78496	0.222376	0.111188	+	0.993799i	$0.464534\pi$
$464$	0	0
$465$	−66.8839	−3.10167
$466$	0	0
$467$	37.6881	1.74400	0.871999	−	0.489508i	$-0.162824\pi$
$467$	37.6881	1.74400	0.871999	+	0.489508i	$0.162824\pi$
$468$	0	0
$469$	40.9719	1.89190
$470$	0	0
$471$	29.3191	1.35095
$472$	0	0
$473$	−9.65009	−0.443711
$474$	0	0
$475$	−95.6557	−4.38899
$476$	0	0
$477$	24.9635	1.14300
$478$	0	0
$479$	−2.57745	−0.117767	−0.0588834	−	0.998265i	$-0.518754\pi$
$479$	−2.57745	−0.117767	−0.0588834	+	0.998265i	$0.518754\pi$
$480$	0	0
$481$	20.0523	0.914306
$482$	0	0
$483$	−3.77431	−0.171737
$484$	0	0
$485$	17.2821	0.784741
$486$	0	0
$487$	4.99998	0.226571	0.113285	−	0.993562i	$-0.463863\pi$
$487$	4.99998	0.226571	0.113285	+	0.993562i	$0.463863\pi$
$488$	0	0
$489$	11.2692	0.509612
$490$	0	0
$491$	16.9889	0.766699	0.383350	−	0.923603i	$-0.374770\pi$
$491$	16.9889	0.766699	0.383350	+	0.923603i	$0.374770\pi$
$492$	0	0
$493$	−2.21948	−0.0999602
$494$	0	0
$495$	45.3428	2.03801
$496$	0	0
$497$	51.3731	2.30440
$498$	0	0
$499$	13.9474	0.624373	0.312187	−	0.950021i	$-0.398939\pi$
$499$	13.9474	0.624373	0.312187	+	0.950021i	$0.398939\pi$
$500$	0	0
$501$	−43.3880	−1.93843
$502$	0	0
$503$	−21.7036	−0.967717	−0.483858	−	0.875146i	$-0.660765\pi$
$503$	−21.7036	−0.967717	−0.483858	+	0.875146i	$0.660765\pi$
$504$	0	0
$505$	52.6832	2.34437
$506$	0	0
$507$	12.3995	0.550682
$508$	0	0
$509$	−21.7333	−0.963311	−0.481656	−	0.876361i	$-0.659964\pi$
$509$	−21.7333	−0.963311	−0.481656	+	0.876361i	$0.659964\pi$
$510$	0	0
$511$	32.7719	1.44975
$512$	0	0
$513$	41.4190	1.82869
$514$	0	0
$515$	77.3268	3.40743
$516$	0	0
$517$	−19.0918	−0.839657
$518$	0	0
$519$	−17.6412	−0.774363
$520$	0	0
$521$	17.0548	0.747186	0.373593	−	0.927593i	$-0.378126\pi$
$521$	17.0548	0.747186	0.373593	+	0.927593i	$0.378126\pi$
$522$	0	0
$523$	41.3195	1.80678	0.903388	−	0.428824i	$-0.141072\pi$
$523$	41.3195	1.80678	0.903388	+	0.428824i	$0.141072\pi$
$524$	0	0
$525$	−145.628	−6.35572
$526$	0	0
$527$	−15.7543	−0.686266
$528$	0	0
$529$	−22.8832	−0.994920
$530$	0	0
$531$	7.66636	0.332692
$532$	0	0
$533$	−26.0181	−1.12697
$534$	0	0
$535$	−76.1003	−3.29010
$536$	0	0
$537$	−4.81283	−0.207689
$538$	0	0
$539$	17.3939	0.749208
$540$	0	0
$541$	−38.4177	−1.65170	−0.825852	−	0.563887i	$-0.809305\pi$
$541$	−38.4177	−1.65170	−0.825852	+	0.563887i	$0.809305\pi$
$542$	0	0
$543$	38.4394	1.64959
$544$	0	0
$545$	72.8448	3.12033
$546$	0	0
$547$	−34.4323	−1.47222	−0.736109	−	0.676864i	$-0.763339\pi$
$547$	−34.4323	−1.47222	−0.736109	+	0.676864i	$0.763339\pi$
$548$	0	0
$549$	40.0115	1.70765
$550$	0	0
$551$	−5.65942	−0.241099
$552$	0	0
$553$	−9.31111	−0.395949
$554$	0	0
$555$	−82.4704	−3.50068
$556$	0	0
$557$	−31.8370	−1.34898	−0.674488	−	0.738286i	$-0.735636\pi$
$557$	−31.8370	−1.34898	−0.674488	+	0.738286i	$0.735636\pi$
$558$	0	0
$559$	−13.3701	−0.565496
$560$	0	0
$561$	17.0669	0.720565
$562$	0	0
$563$	−11.4508	−0.482592	−0.241296	−	0.970452i	$-0.577572\pi$
$563$	−11.4508	−0.482592	−0.241296	+	0.970452i	$0.577572\pi$
$564$	0	0
$565$	−39.8576	−1.67682
$566$	0	0
$567$	4.36308	0.183232
$568$	0	0
$569$	−25.5735	−1.07210	−0.536049	−	0.844187i	$-0.680084\pi$
$569$	−25.5735	−1.07210	−0.536049	+	0.844187i	$0.680084\pi$
$570$	0	0
$571$	6.15130	0.257424	0.128712	−	0.991682i	$-0.458916\pi$
$571$	6.15130	0.257424	0.128712	+	0.991682i	$0.458916\pi$
$572$	0	0
$573$	41.2082	1.72150
$574$	0	0
$575$	4.50824	0.188007
$576$	0	0
$577$	−17.0654	−0.710444	−0.355222	−	0.934782i	$-0.615595\pi$
$577$	−17.0654	−0.710444	−0.355222	+	0.934782i	$0.615595\pi$
$578$	0	0
$579$	70.6448	2.93590
$580$	0	0
$581$	−39.2821	−1.62969
$582$	0	0
$583$	10.5447	0.436717
$584$	0	0
$585$	62.8222	2.59738
$586$	0	0
$587$	26.9333	1.11166	0.555828	−	0.831298i	$-0.312401\pi$
$587$	26.9333	1.11166	0.555828	+	0.831298i	$0.312401\pi$
$588$	0	0
$589$	−40.1716	−1.65524
$590$	0	0
$591$	53.8437	2.21484
$592$	0	0
$593$	−24.1893	−0.993334	−0.496667	−	0.867941i	$-0.665443\pi$
$593$	−24.1893	−0.993334	−0.496667	+	0.867941i	$0.665443\pi$
$594$	0	0
$595$	−47.3063	−1.93937
$596$	0	0
$597$	−56.4204	−2.30913
$598$	0	0
$599$	−7.86196	−0.321231	−0.160615	−	0.987017i	$-0.551348\pi$
$599$	−7.86196	−0.321231	−0.160615	+	0.987017i	$0.551348\pi$
$600$	0	0
$601$	−11.6031	−0.473300	−0.236650	−	0.971595i	$-0.576049\pi$
$601$	−11.6031	−0.473300	−0.236650	+	0.971595i	$0.576049\pi$
$602$	0	0
$603$	52.7097	2.14650
$604$	0	0
$605$	−27.7602	−1.12861
$606$	0	0
$607$	−16.6568	−0.676080	−0.338040	−	0.941132i	$-0.609764\pi$
$607$	−16.6568	−0.676080	−0.338040	+	0.941132i	$0.609764\pi$
$608$	0	0
$609$	−8.61600	−0.349138
$610$	0	0
$611$	−26.4516	−1.07012
$612$	0	0
$613$	9.70838	0.392118	0.196059	−	0.980592i	$-0.437186\pi$
$613$	9.70838	0.392118	0.196059	+	0.980592i	$0.437186\pi$
$614$	0	0
$615$	107.007	4.31492
$616$	0	0
$617$	24.9271	1.00353	0.501764	−	0.865004i	$-0.332684\pi$
$617$	24.9271	1.00353	0.501764	+	0.865004i	$0.332684\pi$
$618$	0	0
$619$	−24.2158	−0.973316	−0.486658	−	0.873593i	$-0.661784\pi$
$619$	−24.2158	−0.973316	−0.486658	+	0.873593i	$0.661784\pi$
$620$	0	0
$621$	−1.95207	−0.0783339
$622$	0	0
$623$	15.2636	0.611524
$624$	0	0
$625$	83.0015	3.32006
$626$	0	0
$627$	43.5187	1.73797
$628$	0	0
$629$	−19.4256	−0.774549
$630$	0	0
$631$	9.57832	0.381307	0.190653	−	0.981657i	$-0.438939\pi$
$631$	9.57832	0.381307	0.190653	+	0.981657i	$0.438939\pi$
$632$	0	0
$633$	−67.9400	−2.70037
$634$	0	0
$635$	−69.4002	−2.75406
$636$	0	0
$637$	24.0991	0.954841
$638$	0	0
$639$	66.0907	2.61451
$640$	0	0
$641$	−30.2024	−1.19292	−0.596461	−	0.802642i	$-0.703427\pi$
$641$	−30.2024	−1.19292	−0.596461	+	0.802642i	$0.703427\pi$
$642$	0	0
$643$	−29.2715	−1.15436	−0.577178	−	0.816618i	$-0.695846\pi$
$643$	−29.2715	−1.15436	−0.577178	+	0.816618i	$0.695846\pi$
$644$	0	0
$645$	54.9883	2.16516
$646$	0	0
$647$	27.8158	1.09355	0.546776	−	0.837279i	$-0.315855\pi$
$647$	27.8158	1.09355	0.546776	+	0.837279i	$0.315855\pi$
$648$	0	0
$649$	3.23831	0.127115
$650$	0	0
$651$	−61.1579	−2.39697
$652$	0	0
$653$	−5.73215	−0.224316	−0.112158	−	0.993690i	$-0.535776\pi$
$653$	−5.73215	−0.224316	−0.112158	+	0.993690i	$0.535776\pi$
$654$	0	0
$655$	0.0790439	0.00308850
$656$	0	0
$657$	42.1606	1.64484
$658$	0	0
$659$	32.2628	1.25678	0.628390	−	0.777898i	$-0.283714\pi$
$659$	32.2628	1.25678	0.628390	+	0.777898i	$0.283714\pi$
$660$	0	0
$661$	−12.2122	−0.474999	−0.237500	−	0.971388i	$-0.576328\pi$
$661$	−12.2122	−0.474999	−0.237500	+	0.971388i	$0.576328\pi$
$662$	0	0
$663$	23.6461	0.918338
$664$	0	0
$665$	−120.626	−4.67767
$666$	0	0
$667$	0.266728	0.0103277
$668$	0	0
$669$	−23.8960	−0.923874
$670$	0	0
$671$	16.9011	0.652459
$672$	0	0
$673$	4.21631	0.162527	0.0812635	−	0.996693i	$-0.474104\pi$
$673$	4.21631	0.162527	0.0812635	+	0.996693i	$0.474104\pi$
$674$	0	0
$675$	−75.3186	−2.89902
$676$	0	0
$677$	37.3291	1.43467	0.717337	−	0.696726i	$-0.245361\pi$
$677$	37.3291	1.43467	0.717337	+	0.696726i	$0.245361\pi$
$678$	0	0
$679$	15.8026	0.606448
$680$	0	0
$681$	−15.6282	−0.598873
$682$	0	0
$683$	28.4120	1.08715	0.543577	−	0.839359i	$-0.317069\pi$
$683$	28.4120	1.08715	0.543577	+	0.839359i	$0.317069\pi$
$684$	0	0
$685$	8.26676	0.315857
$686$	0	0
$687$	−47.6522	−1.81804
$688$	0	0
$689$	14.6096	0.556582
$690$	0	0
$691$	−32.8252	−1.24873	−0.624364	−	0.781134i	$-0.714642\pi$
$691$	−32.8252	−1.24873	−0.624364	+	0.781134i	$0.714642\pi$
$692$	0	0
$693$	41.4610	1.57497
$694$	0	0
$695$	−49.6423	−1.88304
$696$	0	0
$697$	25.2050	0.954707
$698$	0	0
$699$	15.9503	0.603295
$700$	0	0
$701$	−31.2548	−1.18048	−0.590239	−	0.807229i	$-0.700967\pi$
$701$	−31.2548	−1.18048	−0.590239	+	0.807229i	$0.700967\pi$
$702$	0	0
$703$	−49.5331	−1.86818
$704$	0	0
$705$	108.789	4.09724
$706$	0	0
$707$	48.1729	1.81173
$708$	0	0
$709$	−2.69132	−0.101075	−0.0505373	−	0.998722i	$-0.516093\pi$
$709$	−2.69132	−0.101075	−0.0505373	+	0.998722i	$0.516093\pi$
$710$	0	0
$711$	−11.9786	−0.449233
$712$	0	0
$713$	1.89328	0.0709039
$714$	0	0
$715$	26.5364	0.992407
$716$	0	0
$717$	65.9309	2.46223
$718$	0	0
$719$	−5.33934	−0.199124	−0.0995620	−	0.995031i	$-0.531744\pi$
$719$	−5.33934	−0.199124	−0.0995620	+	0.995031i	$0.531744\pi$
$720$	0	0
$721$	70.7068	2.63326
$722$	0	0
$723$	63.8870	2.37598
$724$	0	0
$725$	10.2914	0.382213
$726$	0	0
$727$	−13.9052	−0.515716	−0.257858	−	0.966183i	$-0.583017\pi$
$727$	−13.9052	−0.515716	−0.257858	+	0.966183i	$0.583017\pi$
$728$	0	0
$729$	−42.8958	−1.58873
$730$	0	0
$731$	12.9523	0.479057
$732$	0	0
$733$	10.5028	0.387931	0.193965	−	0.981008i	$-0.437865\pi$
$733$	10.5028	0.387931	0.193965	+	0.981008i	$0.437865\pi$
$734$	0	0
$735$	−99.1141	−3.65588
$736$	0	0
$737$	22.2649	0.820136
$738$	0	0
$739$	−31.6931	−1.16585	−0.582925	−	0.812526i	$-0.698092\pi$
$739$	−31.6931	−1.16585	−0.582925	+	0.812526i	$0.698092\pi$
$740$	0	0
$741$	60.2949	2.21499
$742$	0	0
$743$	9.02988	0.331274	0.165637	−	0.986187i	$-0.447032\pi$
$743$	9.02988	0.331274	0.165637	+	0.986187i	$0.447032\pi$
$744$	0	0
$745$	9.90386	0.362849
$746$	0	0
$747$	−50.5358	−1.84901
$748$	0	0
$749$	−69.5853	−2.54259
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−41.2671	−1.50386
$754$	0	0
$755$	−46.5341	−1.69355
$756$	0	0
$757$	−16.5770	−0.602503	−0.301251	−	0.953545i	$-0.597404\pi$
$757$	−16.5770	−0.602503	−0.301251	+	0.953545i	$0.597404\pi$
$758$	0	0
$759$	−2.05103	−0.0744477
$760$	0	0
$761$	25.0104	0.906627	0.453313	−	0.891351i	$-0.350242\pi$
$761$	25.0104	0.906627	0.453313	+	0.891351i	$0.350242\pi$
$762$	0	0
$763$	66.6085	2.41139
$764$	0	0
$765$	−60.8588	−2.20036
$766$	0	0
$767$	4.48666	0.162004
$768$	0	0
$769$	26.5552	0.957606	0.478803	−	0.877922i	$-0.341071\pi$
$769$	26.5552	0.957606	0.478803	+	0.877922i	$0.341071\pi$
$770$	0	0
$771$	−35.6234	−1.28294
$772$	0	0
$773$	−6.05848	−0.217908	−0.108954	−	0.994047i	$-0.534750\pi$
$773$	−6.05848	−0.217908	−0.108954	+	0.994047i	$0.534750\pi$
$774$	0	0
$775$	73.0503	2.62404
$776$	0	0
$777$	−75.4101	−2.70532
$778$	0	0
$779$	64.2700	2.30271
$780$	0	0
$781$	27.9171	0.998952
$782$	0	0
$783$	−4.45619	−0.159251
$784$	0	0
$785$	−44.1621	−1.57621
$786$	0	0
$787$	5.26096	0.187533	0.0937665	−	0.995594i	$-0.470109\pi$
$787$	5.26096	0.187533	0.0937665	+	0.995594i	$0.470109\pi$
$788$	0	0
$789$	13.0467	0.464476
$790$	0	0
$791$	−36.4454	−1.29585
$792$	0	0
$793$	23.4163	0.831538
$794$	0	0
$795$	−60.0860	−2.13103
$796$	0	0
$797$	4.80092	0.170057	0.0850287	−	0.996378i	$-0.472902\pi$
$797$	4.80092	0.170057	0.0850287	+	0.996378i	$0.472902\pi$
$798$	0	0
$799$	25.6249	0.906544
$800$	0	0
$801$	19.6364	0.693819
$802$	0	0
$803$	17.8089	0.628461
$804$	0	0
$805$	5.68508	0.200373
$806$	0	0
$807$	−32.6789	−1.15035
$808$	0	0
$809$	−38.2087	−1.34335	−0.671674	−	0.740847i	$-0.734424\pi$
$809$	−38.2087	−1.34335	−0.671674	+	0.740847i	$0.734424\pi$
$810$	0	0
$811$	30.7417	1.07949	0.539743	−	0.841830i	$-0.318521\pi$
$811$	30.7417	1.07949	0.539743	+	0.841830i	$0.318521\pi$
$812$	0	0
$813$	−83.8577	−2.94102
$814$	0	0
$815$	−16.9743	−0.594584
$816$	0	0
$817$	33.0269	1.15546
$818$	0	0
$819$	57.4439	2.00725
$820$	0	0
$821$	17.2126	0.600725	0.300363	−	0.953825i	$-0.402892\pi$
$821$	17.2126	0.600725	0.300363	+	0.953825i	$0.402892\pi$
$822$	0	0
$823$	−7.02994	−0.245048	−0.122524	−	0.992466i	$-0.539099\pi$
$823$	−7.02994	−0.245048	−0.122524	+	0.992466i	$0.539099\pi$
$824$	0	0
$825$	−79.1369	−2.75519
$826$	0	0
$827$	46.4072	1.61374	0.806869	−	0.590731i	$-0.201161\pi$
$827$	46.4072	1.61374	0.806869	+	0.590731i	$0.201161\pi$
$828$	0	0
$829$	−45.7914	−1.59040	−0.795201	−	0.606346i	$-0.792635\pi$
$829$	−45.7914	−1.59040	−0.795201	+	0.606346i	$0.792635\pi$
$830$	0	0
$831$	−79.8541	−2.77011
$832$	0	0
$833$	−23.3459	−0.808889
$834$	0	0
$835$	65.3534	2.26165
$836$	0	0
$837$	−31.6308	−1.09332
$838$	0	0
$839$	−7.11008	−0.245467	−0.122734	−	0.992440i	$-0.539166\pi$
$839$	−7.11008	−0.245467	−0.122734	+	0.992440i	$0.539166\pi$
$840$	0	0
$841$	−28.3911	−0.979004
$842$	0	0
$843$	−4.41226	−0.151966
$844$	0	0
$845$	−18.6768	−0.642503
$846$	0	0
$847$	−25.3836	−0.872190
$848$	0	0
$849$	10.2353	0.351273
$850$	0	0
$851$	2.33449	0.0800252
$852$	0	0
$853$	19.4543	0.666102	0.333051	−	0.942909i	$-0.391922\pi$
$853$	19.4543	0.666102	0.333051	+	0.942909i	$0.391922\pi$
$854$	0	0
$855$	−155.183	−5.30716
$856$	0	0
$857$	−21.4375	−0.732290	−0.366145	−	0.930558i	$-0.619323\pi$
$857$	−21.4375	−0.732290	−0.366145	+	0.930558i	$0.619323\pi$
$858$	0	0
$859$	−30.0759	−1.02618	−0.513089	−	0.858335i	$-0.671499\pi$
$859$	−30.0759	−1.02618	−0.513089	+	0.858335i	$0.671499\pi$
$860$	0	0
$861$	97.8457	3.33457
$862$	0	0
$863$	24.2949	0.827008	0.413504	−	0.910502i	$-0.364305\pi$
$863$	24.2949	0.827008	0.413504	+	0.910502i	$0.364305\pi$
$864$	0	0
$865$	26.5722	0.903480
$866$	0	0
$867$	25.2270	0.856755
$868$	0	0
$869$	−5.05983	−0.171643
$870$	0	0
$871$	30.8478	1.04524
$872$	0	0
$873$	20.3298	0.688059
$874$	0	0
$875$	136.194	4.60421
$876$	0	0
$877$	31.2901	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$877$	31.2901	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$878$	0	0
$879$	−44.4108	−1.49794
$880$	0	0
$881$	1.89954	0.0639970	0.0319985	−	0.999488i	$-0.489813\pi$
$881$	1.89954	0.0639970	0.0319985	+	0.999488i	$0.489813\pi$
$882$	0	0
$883$	55.1920	1.85736	0.928679	−	0.370884i	$-0.120945\pi$
$883$	55.1920	1.85736	0.928679	+	0.370884i	$0.120945\pi$
$884$	0	0
$885$	−18.4526	−0.620278
$886$	0	0
$887$	−29.6398	−0.995206	−0.497603	−	0.867405i	$-0.665786\pi$
$887$	−29.6398	−0.995206	−0.497603	+	0.867405i	$0.665786\pi$
$888$	0	0
$889$	−63.4588	−2.12834
$890$	0	0
$891$	2.37098	0.0794307
$892$	0	0
$893$	65.3407	2.18654
$894$	0	0
$895$	7.24935	0.242319
$896$	0	0
$897$	−2.84169	−0.0948812
$898$	0	0
$899$	4.32198	0.144146
$900$	0	0
$901$	−14.1530	−0.471506
$902$	0	0
$903$	50.2807	1.67324
$904$	0	0
$905$	−57.8995	−1.92465
$906$	0	0
$907$	48.0318	1.59487	0.797436	−	0.603404i	$-0.206189\pi$
$907$	48.0318	1.59487	0.797436	+	0.603404i	$0.206189\pi$
$908$	0	0
$909$	61.9737	2.05554
$910$	0	0
$911$	−20.2868	−0.672133	−0.336066	−	0.941838i	$-0.609097\pi$
$911$	−20.2868	−0.672133	−0.336066	+	0.941838i	$0.609097\pi$
$912$	0	0
$913$	−21.3466	−0.706469
$914$	0	0
$915$	−96.3060	−3.18378
$916$	0	0
$917$	0.0722769	0.00238679
$918$	0	0
$919$	20.4138	0.673389	0.336694	−	0.941614i	$-0.390691\pi$
$919$	20.4138	0.673389	0.336694	+	0.941614i	$0.390691\pi$
$920$	0	0
$921$	−72.1490	−2.37739
$922$	0	0
$923$	38.6789	1.27313
$924$	0	0
$925$	90.0738	2.96161
$926$	0	0
$927$	90.9632	2.98762
$928$	0	0
$929$	15.9611	0.523666	0.261833	−	0.965113i	$-0.415673\pi$
$929$	15.9611	0.523666	0.261833	+	0.965113i	$0.415673\pi$
$930$	0	0
$931$	−59.5296	−1.95100
$932$	0	0
$933$	−75.8568	−2.48344
$934$	0	0
$935$	−25.7071	−0.840712
$936$	0	0
$937$	21.6107	0.705991	0.352996	−	0.935625i	$-0.385163\pi$
$937$	21.6107	0.705991	0.352996	+	0.935625i	$0.385163\pi$
$938$	0	0
$939$	−92.4261	−3.01621
$940$	0	0
$941$	27.0829	0.882876	0.441438	−	0.897292i	$-0.354469\pi$
$941$	27.0829	0.882876	0.441438	+	0.897292i	$0.354469\pi$
$942$	0	0
$943$	−3.02903	−0.0986389
$944$	0	0
$945$	−94.9799	−3.08970
$946$	0	0
$947$	51.6301	1.67775	0.838876	−	0.544323i	$-0.183213\pi$
$947$	51.6301	1.67775	0.838876	+	0.544323i	$0.183213\pi$
$948$	0	0
$949$	24.6741	0.800954
$950$	0	0
$951$	−14.8766	−0.482408
$952$	0	0
$953$	−58.6944	−1.90130	−0.950649	−	0.310270i	$-0.899581\pi$
$953$	−58.6944	−1.90130	−0.950649	+	0.310270i	$0.899581\pi$
$954$	0	0
$955$	−62.0701	−2.00854
$956$	0	0
$957$	−4.68209	−0.151350
$958$	0	0
$959$	7.55903	0.244094
$960$	0	0
$961$	−0.321783	−0.0103801
$962$	0	0
$963$	−89.5205	−2.88476
$964$	0	0
$965$	−106.409	−3.42543
$966$	0	0
$967$	40.5109	1.30274	0.651371	−	0.758759i	$-0.274194\pi$
$967$	40.5109	1.30274	0.651371	+	0.758759i	$0.274194\pi$
$968$	0	0
$969$	−58.4105	−1.87642
$970$	0	0
$971$	12.3299	0.395686	0.197843	−	0.980234i	$-0.436606\pi$
$971$	12.3299	0.395686	0.197843	+	0.980234i	$0.436606\pi$
$972$	0	0
$973$	−45.3924	−1.45521
$974$	0	0
$975$	−109.644	−3.51141
$976$	0	0
$977$	19.6190	0.627666	0.313833	−	0.949478i	$-0.398387\pi$
$977$	19.6190	0.627666	0.313833	+	0.949478i	$0.398387\pi$
$978$	0	0
$979$	8.29453	0.265094
$980$	0	0
$981$	85.6908	2.73590
$982$	0	0
$983$	16.6017	0.529512	0.264756	−	0.964315i	$-0.414709\pi$
$983$	16.6017	0.529512	0.264756	+	0.964315i	$0.414709\pi$
$984$	0	0
$985$	−81.1024	−2.58414
$986$	0	0
$987$	99.4757	3.16635
$988$	0	0
$989$	−1.55655	−0.0494954
$990$	0	0
$991$	12.4425	0.395249	0.197624	−	0.980278i	$-0.436677\pi$
$991$	12.4425	0.395249	0.197624	+	0.980278i	$0.436677\pi$
$992$	0	0
$993$	59.1346	1.87658
$994$	0	0
$995$	84.9835	2.69416
$996$	0	0
$997$	−30.4078	−0.963023	−0.481512	−	0.876440i	$-0.659912\pi$
$997$	−30.4078	−0.963023	−0.481512	+	0.876440i	$0.659912\pi$
$998$	0	0
$999$	−39.0020	−1.23397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.5	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-2.83142 q^{3} +4.26484 q^{5} +3.89972 q^{7} +5.01693 q^{9} +O(q^{10})\)	\(q-2.83142 q^{3} +4.26484 q^{5} +3.89972 q^{7} +5.01693 q^{9} +2.11918 q^{11} +2.93611 q^{13} -12.0755 q^{15} -2.84435 q^{17} -7.25277 q^{19} -11.0417 q^{21} +0.341822 q^{23} +13.1888 q^{25} -5.71078 q^{27} +0.780311 q^{29} +5.53879 q^{31} -6.00029 q^{33} +16.6317 q^{35} +6.82954 q^{37} -8.31336 q^{39} -8.86143 q^{41} -4.55369 q^{43} +21.3964 q^{45} -9.00906 q^{47} +8.20784 q^{49} +8.05354 q^{51} +4.97585 q^{53} +9.03796 q^{55} +20.5356 q^{57} +1.52810 q^{59} +7.97529 q^{61} +19.5646 q^{63} +12.5220 q^{65} +10.5064 q^{67} -0.967841 q^{69} +13.1735 q^{71} +8.40366 q^{73} -37.3432 q^{75} +8.26421 q^{77} -2.38763 q^{79} +1.11882 q^{81} -10.0730 q^{83} -12.1307 q^{85} -2.20939 q^{87} +3.91403 q^{89} +11.4500 q^{91} -15.6826 q^{93} -30.9319 q^{95} +4.05224 q^{97} +10.6318 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more