LMFDB - Embedded newform 6008.2.a.e.1.49

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			1.49
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)

$f(q)$	$=$	$q+3.31674 q^{3} +1.75910 q^{5} +2.38162 q^{7} +8.00078 q^{9} +O(q^{10})$	$q+3.31674 q^{3} +1.75910 q^{5} +2.38162 q^{7} +8.00078 q^{9} -3.93150 q^{11} -0.976931 q^{13} +5.83449 q^{15} -2.10948 q^{17} +6.21202 q^{19} +7.89923 q^{21} +6.40852 q^{23} -1.90555 q^{25} +16.5863 q^{27} +0.867057 q^{29} -10.0166 q^{31} -13.0398 q^{33} +4.18952 q^{35} +0.819052 q^{37} -3.24023 q^{39} +4.02372 q^{41} +6.72951 q^{43} +14.0742 q^{45} +8.59758 q^{47} -1.32788 q^{49} -6.99660 q^{51} +2.10552 q^{53} -6.91592 q^{55} +20.6037 q^{57} -5.86417 q^{59} -10.1195 q^{61} +19.0548 q^{63} -1.71852 q^{65} -3.94234 q^{67} +21.2554 q^{69} +3.41828 q^{71} -7.00064 q^{73} -6.32024 q^{75} -9.36335 q^{77} -0.747192 q^{79} +31.0102 q^{81} -1.07637 q^{83} -3.71079 q^{85} +2.87580 q^{87} +0.430535 q^{89} -2.32668 q^{91} -33.2225 q^{93} +10.9276 q^{95} +8.89693 q^{97} -31.4551 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.31674	1.91492	0.957461	−	0.288562i	$-0.0931772\pi$
$3$	3.31674	1.91492	0.957461	+	0.288562i	$0.0931772\pi$
$4$	0	0
$5$	1.75910	0.786695	0.393347	−	0.919390i	$-0.371317\pi$
$5$	1.75910	0.786695	0.393347	+	0.919390i	$0.371317\pi$
$6$	0	0
$7$	2.38162	0.900169	0.450084	−	0.892986i	$-0.351394\pi$
$7$	2.38162	0.900169	0.450084	+	0.892986i	$0.351394\pi$
$8$	0	0
$9$	8.00078	2.66693
$10$	0	0
$11$	−3.93150	−1.18539	−0.592696	−	0.805426i	$-0.701937\pi$
$11$	−3.93150	−1.18539	−0.592696	+	0.805426i	$0.701937\pi$
$12$	0	0
$13$	−0.976931	−0.270952	−0.135476	−	0.990781i	$-0.543256\pi$
$13$	−0.976931	−0.270952	−0.135476	+	0.990781i	$0.543256\pi$
$14$	0	0
$15$	5.83449	1.50646
$16$	0	0
$17$	−2.10948	−0.511624	−0.255812	−	0.966727i	$-0.582343\pi$
$17$	−2.10948	−0.511624	−0.255812	+	0.966727i	$0.582343\pi$
$18$	0	0
$19$	6.21202	1.42513	0.712567	−	0.701604i	$-0.247532\pi$
$19$	6.21202	1.42513	0.712567	+	0.701604i	$0.247532\pi$
$20$	0	0
$21$	7.89923	1.72375
$22$	0	0
$23$	6.40852	1.33627	0.668135	−	0.744040i	$-0.267093\pi$
$23$	6.40852	1.33627	0.668135	+	0.744040i	$0.267093\pi$
$24$	0	0
$25$	−1.90555	−0.381111
$26$	0	0
$27$	16.5863	3.19204
$28$	0	0
$29$	0.867057	0.161008	0.0805042	−	0.996754i	$-0.474347\pi$
$29$	0.867057	0.161008	0.0805042	+	0.996754i	$0.474347\pi$
$30$	0	0
$31$	−10.0166	−1.79903	−0.899517	−	0.436885i	$-0.856082\pi$
$31$	−10.0166	−1.79903	−0.899517	+	0.436885i	$0.856082\pi$
$32$	0	0
$33$	−13.0398	−2.26993
$34$	0	0
$35$	4.18952	0.708158
$36$	0	0
$37$	0.819052	0.134651	0.0673257	−	0.997731i	$-0.478553\pi$
$37$	0.819052	0.134651	0.0673257	+	0.997731i	$0.478553\pi$
$38$	0	0
$39$	−3.24023	−0.518852
$40$	0	0
$41$	4.02372	0.628399	0.314200	−	0.949357i	$-0.398264\pi$
$41$	4.02372	0.628399	0.314200	+	0.949357i	$0.398264\pi$
$42$	0	0
$43$	6.72951	1.02624	0.513121	−	0.858317i	$-0.328489\pi$
$43$	6.72951	1.02624	0.513121	+	0.858317i	$0.328489\pi$
$44$	0	0
$45$	14.0742	2.09806
$46$	0	0
$47$	8.59758	1.25409	0.627043	−	0.778985i	$-0.284265\pi$
$47$	8.59758	1.25409	0.627043	+	0.778985i	$0.284265\pi$
$48$	0	0
$49$	−1.32788	−0.189696
$50$	0	0
$51$	−6.99660	−0.979720
$52$	0	0
$53$	2.10552	0.289215	0.144608	−	0.989489i	$-0.453808\pi$
$53$	2.10552	0.289215	0.144608	+	0.989489i	$0.453808\pi$
$54$	0	0
$55$	−6.91592	−0.932542
$56$	0	0
$57$	20.6037	2.72902
$58$	0	0
$59$	−5.86417	−0.763450	−0.381725	−	0.924276i	$-0.624670\pi$
$59$	−5.86417	−0.763450	−0.381725	+	0.924276i	$0.624670\pi$
$60$	0	0
$61$	−10.1195	−1.29567	−0.647837	−	0.761779i	$-0.724326\pi$
$61$	−10.1195	−1.29567	−0.647837	+	0.761779i	$0.724326\pi$
$62$	0	0
$63$	19.0548	2.40068
$64$	0	0
$65$	−1.71852	−0.213156
$66$	0	0
$67$	−3.94234	−0.481634	−0.240817	−	0.970571i	$-0.577415\pi$
$67$	−3.94234	−0.481634	−0.240817	+	0.970571i	$0.577415\pi$
$68$	0	0
$69$	21.2554	2.55885
$70$	0	0
$71$	3.41828	0.405675	0.202838	−	0.979212i	$-0.434984\pi$
$71$	3.41828	0.405675	0.202838	+	0.979212i	$0.434984\pi$
$72$	0	0
$73$	−7.00064	−0.819363	−0.409682	−	0.912229i	$-0.634360\pi$
$73$	−7.00064	−0.819363	−0.409682	+	0.912229i	$0.634360\pi$
$74$	0	0
$75$	−6.32024	−0.729798
$76$	0	0
$77$	−9.36335	−1.06705
$78$	0	0
$79$	−0.747192	−0.0840656	−0.0420328	−	0.999116i	$-0.513383\pi$
$79$	−0.747192	−0.0840656	−0.0420328	+	0.999116i	$0.513383\pi$
$80$	0	0
$81$	31.0102	3.44557
$82$	0	0
$83$	−1.07637	−0.118147	−0.0590734	−	0.998254i	$-0.518815\pi$
$83$	−1.07637	−0.118147	−0.0590734	+	0.998254i	$0.518815\pi$
$84$	0	0
$85$	−3.71079	−0.402492
$86$	0	0
$87$	2.87580	0.308319
$88$	0	0
$89$	0.430535	0.0456366	0.0228183	−	0.999740i	$-0.492736\pi$
$89$	0.430535	0.0456366	0.0228183	+	0.999740i	$0.492736\pi$
$90$	0	0
$91$	−2.32668	−0.243902
$92$	0	0
$93$	−33.2225	−3.44501
$94$	0	0
$95$	10.9276	1.12115
$96$	0	0
$97$	8.89693	0.903346	0.451673	−	0.892183i	$-0.350827\pi$
$97$	8.89693	0.903346	0.451673	+	0.892183i	$0.350827\pi$
$98$	0	0
$99$	−31.4551	−3.16136
$100$	0	0
$101$	5.66563	0.563752	0.281876	−	0.959451i	$-0.409043\pi$
$101$	5.66563	0.563752	0.281876	+	0.959451i	$0.409043\pi$
$102$	0	0
$103$	−8.05381	−0.793566	−0.396783	−	0.917913i	$-0.629873\pi$
$103$	−8.05381	−0.793566	−0.396783	+	0.917913i	$0.629873\pi$
$104$	0	0
$105$	13.8956	1.35607
$106$	0	0
$107$	−8.64530	−0.835773	−0.417886	−	0.908499i	$-0.637229\pi$
$107$	−8.64530	−0.835773	−0.417886	+	0.908499i	$0.637229\pi$
$108$	0	0
$109$	8.06456	0.772445	0.386222	−	0.922406i	$-0.373780\pi$
$109$	8.06456	0.772445	0.386222	+	0.922406i	$0.373780\pi$
$110$	0	0
$111$	2.71658	0.257847
$112$	0	0
$113$	9.85519	0.927098	0.463549	−	0.886071i	$-0.346576\pi$
$113$	9.85519	0.927098	0.463549	+	0.886071i	$0.346576\pi$
$114$	0	0
$115$	11.2733	1.05124
$116$	0	0
$117$	−7.81621	−0.722609
$118$	0	0
$119$	−5.02398	−0.460548
$120$	0	0
$121$	4.45671	0.405155
$122$	0	0
$123$	13.3456	1.20334
$124$	0	0
$125$	−12.1476	−1.08651
$126$	0	0
$127$	2.75273	0.244265	0.122133	−	0.992514i	$-0.461027\pi$
$127$	2.75273	0.244265	0.122133	+	0.992514i	$0.461027\pi$
$128$	0	0
$129$	22.3201	1.96517
$130$	0	0
$131$	2.96331	0.258905	0.129453	−	0.991586i	$-0.458678\pi$
$131$	2.96331	0.258905	0.129453	+	0.991586i	$0.458678\pi$
$132$	0	0
$133$	14.7947	1.28286
$134$	0	0
$135$	29.1770	2.51116
$136$	0	0
$137$	3.96540	0.338787	0.169393	−	0.985549i	$-0.445819\pi$
$137$	3.96540	0.338787	0.169393	+	0.985549i	$0.445819\pi$
$138$	0	0
$139$	−8.63154	−0.732118	−0.366059	−	0.930592i	$-0.619293\pi$
$139$	−8.63154	−0.732118	−0.366059	+	0.930592i	$0.619293\pi$
$140$	0	0
$141$	28.5160	2.40148
$142$	0	0
$143$	3.84081	0.321184
$144$	0	0
$145$	1.52524	0.126665
$146$	0	0
$147$	−4.40422	−0.363254
$148$	0	0
$149$	−4.71319	−0.386120	−0.193060	−	0.981187i	$-0.561841\pi$
$149$	−4.71319	−0.386120	−0.193060	+	0.981187i	$0.561841\pi$
$150$	0	0
$151$	5.26744	0.428659	0.214329	−	0.976761i	$-0.431243\pi$
$151$	5.26744	0.428659	0.214329	+	0.976761i	$0.431243\pi$
$152$	0	0
$153$	−16.8775	−1.36446
$154$	0	0
$155$	−17.6202	−1.41529
$156$	0	0
$157$	15.1946	1.21266	0.606330	−	0.795213i	$-0.292641\pi$
$157$	15.1946	1.21266	0.606330	+	0.795213i	$0.292641\pi$
$158$	0	0
$159$	6.98347	0.553825
$160$	0	0
$161$	15.2627	1.20287
$162$	0	0
$163$	19.9251	1.56065	0.780325	−	0.625374i	$-0.215054\pi$
$163$	19.9251	1.56065	0.780325	+	0.625374i	$0.215054\pi$
$164$	0	0
$165$	−22.9383	−1.78575
$166$	0	0
$167$	−17.7038	−1.36996	−0.684980	−	0.728562i	$-0.740189\pi$
$167$	−17.7038	−1.36996	−0.684980	+	0.728562i	$0.740189\pi$
$168$	0	0
$169$	−12.0456	−0.926585
$170$	0	0
$171$	49.7010	3.80073
$172$	0	0
$173$	−6.26071	−0.475993	−0.237996	−	0.971266i	$-0.576491\pi$
$173$	−6.26071	−0.475993	−0.237996	+	0.971266i	$0.576491\pi$
$174$	0	0
$175$	−4.53831	−0.343064
$176$	0	0
$177$	−19.4500	−1.46195
$178$	0	0
$179$	−25.1668	−1.88106	−0.940529	−	0.339713i	$-0.889670\pi$
$179$	−25.1668	−1.88106	−0.940529	+	0.339713i	$0.889670\pi$
$180$	0	0
$181$	0.982740	0.0730465	0.0365232	−	0.999333i	$-0.488372\pi$
$181$	0.982740	0.0730465	0.0365232	+	0.999333i	$0.488372\pi$
$182$	0	0
$183$	−33.5639	−2.48111
$184$	0	0
$185$	1.44080	0.105930
$186$	0	0
$187$	8.29342	0.606475
$188$	0	0
$189$	39.5023	2.87337
$190$	0	0
$191$	16.3117	1.18028	0.590138	−	0.807302i	$-0.299073\pi$
$191$	16.3117	1.18028	0.590138	+	0.807302i	$0.299073\pi$
$192$	0	0
$193$	−12.2187	−0.879519	−0.439760	−	0.898116i	$-0.644936\pi$
$193$	−12.2187	−0.879519	−0.439760	+	0.898116i	$0.644936\pi$
$194$	0	0
$195$	−5.69990	−0.408178
$196$	0	0
$197$	−21.1349	−1.50580	−0.752900	−	0.658135i	$-0.771346\pi$
$197$	−21.1349	−1.50580	−0.752900	+	0.658135i	$0.771346\pi$
$198$	0	0
$199$	0.128325	0.00909672	0.00454836	−	0.999990i	$-0.498552\pi$
$199$	0.128325	0.00909672	0.00454836	+	0.999990i	$0.498552\pi$
$200$	0	0
$201$	−13.0757	−0.922292
$202$	0	0
$203$	2.06500	0.144935
$204$	0	0
$205$	7.07814	0.494358
$206$	0	0
$207$	51.2732	3.56373
$208$	0	0
$209$	−24.4226	−1.68934
$210$	0	0
$211$	26.6665	1.83580	0.917898	−	0.396815i	$-0.129885\pi$
$211$	26.6665	1.83580	0.917898	+	0.396815i	$0.129885\pi$
$212$	0	0
$213$	11.3376	0.776837
$214$	0	0
$215$	11.8379	0.807339
$216$	0	0
$217$	−23.8558	−1.61943
$218$	0	0
$219$	−23.2193	−1.56902
$220$	0	0
$221$	2.06082	0.138625
$222$	0	0
$223$	−13.3628	−0.894838	−0.447419	−	0.894324i	$-0.647657\pi$
$223$	−13.3628	−0.894838	−0.447419	+	0.894324i	$0.647657\pi$
$224$	0	0
$225$	−15.2459	−1.01640
$226$	0	0
$227$	−10.6812	−0.708939	−0.354469	−	0.935068i	$-0.615338\pi$
$227$	−10.6812	−0.708939	−0.354469	+	0.935068i	$0.615338\pi$
$228$	0	0
$229$	24.8347	1.64113	0.820563	−	0.571557i	$-0.193660\pi$
$229$	24.8347	1.64113	0.820563	+	0.571557i	$0.193660\pi$
$230$	0	0
$231$	−31.0558	−2.04332
$232$	0	0
$233$	−19.7268	−1.29234	−0.646172	−	0.763192i	$-0.723631\pi$
$233$	−19.7268	−1.29234	−0.646172	+	0.763192i	$0.723631\pi$
$234$	0	0
$235$	15.1240	0.986583
$236$	0	0
$237$	−2.47824	−0.160979
$238$	0	0
$239$	4.32680	0.279878	0.139939	−	0.990160i	$-0.455309\pi$
$239$	4.32680	0.279878	0.139939	+	0.990160i	$0.455309\pi$
$240$	0	0
$241$	−2.28751	−0.147351	−0.0736757	−	0.997282i	$-0.523473\pi$
$241$	−2.28751	−0.147351	−0.0736757	+	0.997282i	$0.523473\pi$
$242$	0	0
$243$	53.0938	3.40597
$244$	0	0
$245$	−2.33587	−0.149233
$246$	0	0
$247$	−6.06871	−0.386143
$248$	0	0
$249$	−3.57004	−0.226242
$250$	0	0
$251$	−6.75842	−0.426588	−0.213294	−	0.976988i	$-0.568419\pi$
$251$	−6.75842	−0.426588	−0.213294	+	0.976988i	$0.568419\pi$
$252$	0	0
$253$	−25.1951	−1.58400
$254$	0	0
$255$	−12.3077	−0.770741
$256$	0	0
$257$	−22.6495	−1.41284	−0.706420	−	0.707793i	$-0.749691\pi$
$257$	−22.6495	−1.41284	−0.706420	+	0.707793i	$0.749691\pi$
$258$	0	0
$259$	1.95067	0.121209
$260$	0	0
$261$	6.93713	0.429398
$262$	0	0
$263$	3.12946	0.192971	0.0964853	−	0.995334i	$-0.469240\pi$
$263$	3.12946	0.192971	0.0964853	+	0.995334i	$0.469240\pi$
$264$	0	0
$265$	3.70383	0.227524
$266$	0	0
$267$	1.42797	0.0873906
$268$	0	0
$269$	−16.6793	−1.01695	−0.508477	−	0.861076i	$-0.669791\pi$
$269$	−16.6793	−1.01695	−0.508477	+	0.861076i	$0.669791\pi$
$270$	0	0
$271$	14.4276	0.876413	0.438207	−	0.898874i	$-0.355614\pi$
$271$	14.4276	0.876413	0.438207	+	0.898874i	$0.355614\pi$
$272$	0	0
$273$	−7.71700	−0.467054
$274$	0	0
$275$	7.49169	0.451766
$276$	0	0
$277$	−10.3553	−0.622190	−0.311095	−	0.950379i	$-0.600696\pi$
$277$	−10.3553	−0.622190	−0.311095	+	0.950379i	$0.600696\pi$
$278$	0	0
$279$	−80.1407	−4.79790
$280$	0	0
$281$	−23.5233	−1.40328	−0.701642	−	0.712530i	$-0.747549\pi$
$281$	−23.5233	−1.40328	−0.701642	+	0.712530i	$0.747549\pi$
$282$	0	0
$283$	−24.4995	−1.45634	−0.728171	−	0.685395i	$-0.759630\pi$
$283$	−24.4995	−1.45634	−0.728171	+	0.685395i	$0.759630\pi$
$284$	0	0
$285$	36.2440	2.14691
$286$	0	0
$287$	9.58298	0.565665
$288$	0	0
$289$	−12.5501	−0.738241
$290$	0	0
$291$	29.5088	1.72984
$292$	0	0
$293$	13.9689	0.816073	0.408036	−	0.912966i	$-0.366214\pi$
$293$	13.9689	0.816073	0.408036	+	0.912966i	$0.366214\pi$
$294$	0	0
$295$	−10.3157	−0.600603
$296$	0	0
$297$	−65.2091	−3.78382
$298$	0	0
$299$	−6.26068	−0.362065
$300$	0	0
$301$	16.0272	0.923790
$302$	0	0
$303$	18.7914	1.07954
$304$	0	0
$305$	−17.8013	−1.01930
$306$	0	0
$307$	−11.0671	−0.631633	−0.315816	−	0.948820i	$-0.602278\pi$
$307$	−11.0671	−0.631633	−0.315816	+	0.948820i	$0.602278\pi$
$308$	0	0
$309$	−26.7124	−1.51962
$310$	0	0
$311$	10.7238	0.608093	0.304047	−	0.952657i	$-0.401662\pi$
$311$	10.7238	0.608093	0.304047	+	0.952657i	$0.401662\pi$
$312$	0	0
$313$	29.8106	1.68500	0.842498	−	0.538699i	$-0.181084\pi$
$313$	29.8106	1.68500	0.842498	+	0.538699i	$0.181084\pi$
$314$	0	0
$315$	33.5194	1.88861
$316$	0	0
$317$	−0.0880663	−0.00494630	−0.00247315	−	0.999997i	$-0.500787\pi$
$317$	−0.0880663	−0.00494630	−0.00247315	+	0.999997i	$0.500787\pi$
$318$	0	0
$319$	−3.40884	−0.190858
$320$	0	0
$321$	−28.6742	−1.60044
$322$	0	0
$323$	−13.1041	−0.729133
$324$	0	0
$325$	1.86160	0.103263
$326$	0	0
$327$	26.7481	1.47917
$328$	0	0
$329$	20.4762	1.12889
$330$	0	0
$331$	−27.5927	−1.51663	−0.758315	−	0.651888i	$-0.773977\pi$
$331$	−27.5927	−1.51663	−0.758315	+	0.651888i	$0.773977\pi$
$332$	0	0
$333$	6.55306	0.359105
$334$	0	0
$335$	−6.93499	−0.378899
$336$	0	0
$337$	17.3623	0.945787	0.472894	−	0.881120i	$-0.343210\pi$
$337$	17.3623	0.945787	0.472894	+	0.881120i	$0.343210\pi$
$338$	0	0
$339$	32.6871	1.77532
$340$	0	0
$341$	39.3803	2.13256
$342$	0	0
$343$	−19.8339	−1.07093
$344$	0	0
$345$	37.3905	2.01304
$346$	0	0
$347$	8.92927	0.479348	0.239674	−	0.970853i	$-0.422959\pi$
$347$	8.92927	0.479348	0.239674	+	0.970853i	$0.422959\pi$
$348$	0	0
$349$	34.3940	1.84107	0.920535	−	0.390660i	$-0.127753\pi$
$349$	34.3940	1.84107	0.920535	+	0.390660i	$0.127753\pi$
$350$	0	0
$351$	−16.2037	−0.864888
$352$	0	0
$353$	−32.0268	−1.70461	−0.852307	−	0.523041i	$-0.824797\pi$
$353$	−32.0268	−1.70461	−0.852307	+	0.523041i	$0.824797\pi$
$354$	0	0
$355$	6.01311	0.319143
$356$	0	0
$357$	−16.6633	−0.881913
$358$	0	0
$359$	−25.6851	−1.35561	−0.677804	−	0.735242i	$-0.737068\pi$
$359$	−25.6851	−1.35561	−0.677804	+	0.735242i	$0.737068\pi$
$360$	0	0
$361$	19.5892	1.03101
$362$	0	0
$363$	14.7818	0.775841
$364$	0	0
$365$	−12.3149	−0.644589
$366$	0	0
$367$	7.51232	0.392140	0.196070	−	0.980590i	$-0.437182\pi$
$367$	7.51232	0.392140	0.196070	+	0.980590i	$0.437182\pi$
$368$	0	0
$369$	32.1929	1.67589
$370$	0	0
$371$	5.01455	0.260343
$372$	0	0
$373$	−5.60492	−0.290212	−0.145106	−	0.989416i	$-0.546352\pi$
$373$	−5.60492	−0.290212	−0.145106	+	0.989416i	$0.546352\pi$
$374$	0	0
$375$	−40.2904	−2.08059
$376$	0	0
$377$	−0.847055	−0.0436255
$378$	0	0
$379$	11.1734	0.573938	0.286969	−	0.957940i	$-0.407352\pi$
$379$	11.1734	0.573938	0.286969	+	0.957940i	$0.407352\pi$
$380$	0	0
$381$	9.13010	0.467749
$382$	0	0
$383$	15.8231	0.808523	0.404261	−	0.914644i	$-0.367529\pi$
$383$	15.8231	0.808523	0.404261	+	0.914644i	$0.367529\pi$
$384$	0	0
$385$	−16.4711	−0.839445
$386$	0	0
$387$	53.8414	2.73691
$388$	0	0
$389$	−21.6769	−1.09906	−0.549532	−	0.835473i	$-0.685194\pi$
$389$	−21.6769	−1.09906	−0.549532	+	0.835473i	$0.685194\pi$
$390$	0	0
$391$	−13.5186	−0.683667
$392$	0	0
$393$	9.82853	0.495784
$394$	0	0
$395$	−1.31439	−0.0661340
$396$	0	0
$397$	14.2200	0.713682	0.356841	−	0.934165i	$-0.383854\pi$
$397$	14.2200	0.713682	0.356841	+	0.934165i	$0.383854\pi$
$398$	0	0
$399$	49.0702	2.45658
$400$	0	0
$401$	28.3284	1.41466	0.707328	−	0.706886i	$-0.249901\pi$
$401$	28.3284	1.41466	0.707328	+	0.706886i	$0.249901\pi$
$402$	0	0
$403$	9.78553	0.487452
$404$	0	0
$405$	54.5501	2.71062
$406$	0	0
$407$	−3.22010	−0.159615
$408$	0	0
$409$	34.2008	1.69112	0.845559	−	0.533881i	$-0.179267\pi$
$409$	34.2008	1.69112	0.845559	+	0.533881i	$0.179267\pi$
$410$	0	0
$411$	13.1522	0.648751
$412$	0	0
$413$	−13.9662	−0.687234
$414$	0	0
$415$	−1.89344	−0.0929455
$416$	0	0
$417$	−28.6286	−1.40195
$418$	0	0
$419$	−28.1099	−1.37326	−0.686630	−	0.727007i	$-0.740911\pi$
$419$	−28.1099	−1.37326	−0.686630	+	0.727007i	$0.740911\pi$
$420$	0	0
$421$	5.21924	0.254370	0.127185	−	0.991879i	$-0.459406\pi$
$421$	5.21924	0.254370	0.127185	+	0.991879i	$0.459406\pi$
$422$	0	0
$423$	68.7874	3.34456
$424$	0	0
$425$	4.01973	0.194985
$426$	0	0
$427$	−24.1009	−1.16633
$428$	0	0
$429$	12.7390	0.615043
$430$	0	0
$431$	20.1671	0.971414	0.485707	−	0.874122i	$-0.338562\pi$
$431$	20.1671	0.971414	0.485707	+	0.874122i	$0.338562\pi$
$432$	0	0
$433$	−4.58909	−0.220538	−0.110269	−	0.993902i	$-0.535171\pi$
$433$	−4.58909	−0.220538	−0.110269	+	0.993902i	$0.535171\pi$
$434$	0	0
$435$	5.05884	0.242553
$436$	0	0
$437$	39.8099	1.90436
$438$	0	0
$439$	−25.5197	−1.21799	−0.608993	−	0.793175i	$-0.708426\pi$
$439$	−25.5197	−1.21799	−0.608993	+	0.793175i	$0.708426\pi$
$440$	0	0
$441$	−10.6240	−0.505907
$442$	0	0
$443$	−36.6426	−1.74094	−0.870472	−	0.492219i	$-0.836186\pi$
$443$	−36.6426	−1.74094	−0.870472	+	0.492219i	$0.836186\pi$
$444$	0	0
$445$	0.757356	0.0359021
$446$	0	0
$447$	−15.6325	−0.739390
$448$	0	0
$449$	1.32124	0.0623534	0.0311767	−	0.999514i	$-0.490075\pi$
$449$	1.32124	0.0623534	0.0311767	+	0.999514i	$0.490075\pi$
$450$	0	0
$451$	−15.8193	−0.744900
$452$	0	0
$453$	17.4708	0.820848
$454$	0	0
$455$	−4.09287	−0.191877
$456$	0	0
$457$	−12.4209	−0.581025	−0.290512	−	0.956871i	$-0.593826\pi$
$457$	−12.4209	−0.581025	−0.290512	+	0.956871i	$0.593826\pi$
$458$	0	0
$459$	−34.9885	−1.63312
$460$	0	0
$461$	10.9915	0.511924	0.255962	−	0.966687i	$-0.417608\pi$
$461$	10.9915	0.511924	0.255962	+	0.966687i	$0.417608\pi$
$462$	0	0
$463$	4.06277	0.188813	0.0944065	−	0.995534i	$-0.469905\pi$
$463$	4.06277	0.188813	0.0944065	+	0.995534i	$0.469905\pi$
$464$	0	0
$465$	−58.4418	−2.71017
$466$	0	0
$467$	18.5621	0.858950	0.429475	−	0.903079i	$-0.358699\pi$
$467$	18.5621	0.858950	0.429475	+	0.903079i	$0.358699\pi$
$468$	0	0
$469$	−9.38917	−0.433552
$470$	0	0
$471$	50.3966	2.32215
$472$	0	0
$473$	−26.4571	−1.21650
$474$	0	0
$475$	−11.8373	−0.543135
$476$	0	0
$477$	16.8458	0.771316
$478$	0	0
$479$	−27.3434	−1.24935	−0.624675	−	0.780885i	$-0.714769\pi$
$479$	−27.3434	−1.24935	−0.624675	+	0.780885i	$0.714769\pi$
$480$	0	0
$481$	−0.800157	−0.0364840
$482$	0	0
$483$	50.6224	2.30340
$484$	0	0
$485$	15.6506	0.710658
$486$	0	0
$487$	26.9825	1.22269	0.611347	−	0.791363i	$-0.290628\pi$
$487$	26.9825	1.22269	0.611347	+	0.791363i	$0.290628\pi$
$488$	0	0
$489$	66.0863	2.98852
$490$	0	0
$491$	−21.7215	−0.980280	−0.490140	−	0.871644i	$-0.663054\pi$
$491$	−21.7215	−0.980280	−0.490140	+	0.871644i	$0.663054\pi$
$492$	0	0
$493$	−1.82904	−0.0823757
$494$	0	0
$495$	−55.3328	−2.48702
$496$	0	0
$497$	8.14106	0.365176
$498$	0	0
$499$	3.91201	0.175126	0.0875629	−	0.996159i	$-0.472092\pi$
$499$	3.91201	0.175126	0.0875629	+	0.996159i	$0.472092\pi$
$500$	0	0
$501$	−58.7189	−2.62337
$502$	0	0
$503$	−31.8132	−1.41848	−0.709240	−	0.704967i	$-0.750962\pi$
$503$	−31.8132	−1.41848	−0.709240	+	0.704967i	$0.750962\pi$
$504$	0	0
$505$	9.96644	0.443501
$506$	0	0
$507$	−39.9522	−1.77434
$508$	0	0
$509$	−17.8562	−0.791462	−0.395731	−	0.918367i	$-0.629509\pi$
$509$	−17.8562	−0.791462	−0.395731	+	0.918367i	$0.629509\pi$
$510$	0	0
$511$	−16.6729	−0.737565
$512$	0	0
$513$	103.034	4.54908
$514$	0	0
$515$	−14.1675	−0.624294
$516$	0	0
$517$	−33.8014	−1.48658
$518$	0	0
$519$	−20.7651	−0.911489
$520$	0	0
$521$	−29.4351	−1.28957	−0.644787	−	0.764363i	$-0.723054\pi$
$521$	−29.4351	−1.28957	−0.644787	+	0.764363i	$0.723054\pi$
$522$	0	0
$523$	16.4946	0.721259	0.360630	−	0.932709i	$-0.382562\pi$
$523$	16.4946	0.721259	0.360630	+	0.932709i	$0.382562\pi$
$524$	0	0
$525$	−15.0524	−0.656941
$526$	0	0
$527$	21.1298	0.920429
$528$	0	0
$529$	18.0692	0.785617
$530$	0	0
$531$	−46.9180	−2.03607
$532$	0	0
$533$	−3.93089	−0.170266
$534$	0	0
$535$	−15.2080	−0.657498
$536$	0	0
$537$	−83.4719	−3.60208
$538$	0	0
$539$	5.22054	0.224865
$540$	0	0
$541$	21.5430	0.926207	0.463103	−	0.886304i	$-0.346736\pi$
$541$	21.5430	0.926207	0.463103	+	0.886304i	$0.346736\pi$
$542$	0	0
$543$	3.25949	0.139878
$544$	0	0
$545$	14.1864	0.607678
$546$	0	0
$547$	−5.82669	−0.249131	−0.124566	−	0.992211i	$-0.539754\pi$
$547$	−5.82669	−0.249131	−0.124566	+	0.992211i	$0.539754\pi$
$548$	0	0
$549$	−80.9642	−3.45547
$550$	0	0
$551$	5.38617	0.229459
$552$	0	0
$553$	−1.77953	−0.0756732
$554$	0	0
$555$	4.77875	0.202847
$556$	0	0
$557$	40.6603	1.72283	0.861416	−	0.507900i	$-0.169578\pi$
$557$	40.6603	1.72283	0.861416	+	0.507900i	$0.169578\pi$
$558$	0	0
$559$	−6.57427	−0.278062
$560$	0	0
$561$	27.5071	1.16135
$562$	0	0
$563$	−23.1065	−0.973824	−0.486912	−	0.873451i	$-0.661877\pi$
$563$	−23.1065	−0.973824	−0.486912	+	0.873451i	$0.661877\pi$
$564$	0	0
$565$	17.3363	0.729343
$566$	0	0
$567$	73.8545	3.10160
$568$	0	0
$569$	−3.53525	−0.148205	−0.0741027	−	0.997251i	$-0.523609\pi$
$569$	−3.53525	−0.148205	−0.0741027	+	0.997251i	$0.523609\pi$
$570$	0	0
$571$	−22.1162	−0.925533	−0.462767	−	0.886480i	$-0.653143\pi$
$571$	−22.1162	−0.925533	−0.462767	+	0.886480i	$0.653143\pi$
$572$	0	0
$573$	54.1019	2.26014
$574$	0	0
$575$	−12.2118	−0.509267
$576$	0	0
$577$	13.8217	0.575404	0.287702	−	0.957720i	$-0.407109\pi$
$577$	13.8217	0.575404	0.287702	+	0.957720i	$0.407109\pi$
$578$	0	0
$579$	−40.5262	−1.68421
$580$	0	0
$581$	−2.56350	−0.106352
$582$	0	0
$583$	−8.27786	−0.342834
$584$	0	0
$585$	−13.7495	−0.568473
$586$	0	0
$587$	6.83707	0.282196	0.141098	−	0.989996i	$-0.454937\pi$
$587$	6.83707	0.282196	0.141098	+	0.989996i	$0.454937\pi$
$588$	0	0
$589$	−62.2233	−2.56387
$590$	0	0
$591$	−70.0991	−2.88349
$592$	0	0
$593$	−38.6406	−1.58678	−0.793390	−	0.608713i	$-0.791686\pi$
$593$	−38.6406	−1.58678	−0.793390	+	0.608713i	$0.791686\pi$
$594$	0	0
$595$	−8.83771	−0.362311
$596$	0	0
$597$	0.425621	0.0174195
$598$	0	0
$599$	1.98783	0.0812206	0.0406103	−	0.999175i	$-0.487070\pi$
$599$	1.98783	0.0812206	0.0406103	+	0.999175i	$0.487070\pi$
$600$	0	0
$601$	1.95243	0.0796411	0.0398206	−	0.999207i	$-0.487321\pi$
$601$	1.95243	0.0796411	0.0398206	+	0.999207i	$0.487321\pi$
$602$	0	0
$603$	−31.5418	−1.28448
$604$	0	0
$605$	7.83981	0.318734
$606$	0	0
$607$	−17.3571	−0.704503	−0.352252	−	0.935905i	$-0.614584\pi$
$607$	−17.3571	−0.704503	−0.352252	+	0.935905i	$0.614584\pi$
$608$	0	0
$609$	6.84908	0.277539
$610$	0	0
$611$	−8.39924	−0.339797
$612$	0	0
$613$	−26.8090	−1.08280	−0.541402	−	0.840764i	$-0.682106\pi$
$613$	−26.8090	−1.08280	−0.541402	+	0.840764i	$0.682106\pi$
$614$	0	0
$615$	23.4764	0.946658
$616$	0	0
$617$	−13.6812	−0.550786	−0.275393	−	0.961332i	$-0.588808\pi$
$617$	−13.6812	−0.550786	−0.275393	+	0.961332i	$0.588808\pi$
$618$	0	0
$619$	−11.8080	−0.474603	−0.237301	−	0.971436i	$-0.576263\pi$
$619$	−11.8080	−0.474603	−0.237301	+	0.971436i	$0.576263\pi$
$620$	0	0
$621$	106.294	4.26542
$622$	0	0
$623$	1.02537	0.0410807
$624$	0	0
$625$	−11.8411	−0.473643
$626$	0	0
$627$	−81.0034	−3.23496
$628$	0	0
$629$	−1.72777	−0.0688908
$630$	0	0
$631$	−23.9873	−0.954920	−0.477460	−	0.878654i	$-0.658442\pi$
$631$	−23.9873	−0.954920	−0.477460	+	0.878654i	$0.658442\pi$
$632$	0	0
$633$	88.4459	3.51541
$634$	0	0
$635$	4.84234	0.192162
$636$	0	0
$637$	1.29724	0.0513986
$638$	0	0
$639$	27.3489	1.08191
$640$	0	0
$641$	−12.5436	−0.495443	−0.247721	−	0.968831i	$-0.579682\pi$
$641$	−12.5436	−0.495443	−0.247721	+	0.968831i	$0.579682\pi$
$642$	0	0
$643$	33.1718	1.30817	0.654085	−	0.756421i	$-0.273054\pi$
$643$	33.1718	1.30817	0.654085	+	0.756421i	$0.273054\pi$
$644$	0	0
$645$	39.2633	1.54599
$646$	0	0
$647$	14.0134	0.550923	0.275461	−	0.961312i	$-0.411169\pi$
$647$	14.0134	0.550923	0.275461	+	0.961312i	$0.411169\pi$
$648$	0	0
$649$	23.0550	0.904988
$650$	0	0
$651$	−79.1234	−3.10109
$652$	0	0
$653$	12.5815	0.492354	0.246177	−	0.969225i	$-0.420826\pi$
$653$	12.5815	0.492354	0.246177	+	0.969225i	$0.420826\pi$
$654$	0	0
$655$	5.21277	0.203680
$656$	0	0
$657$	−56.0106	−2.18518
$658$	0	0
$659$	25.9476	1.01077	0.505387	−	0.862893i	$-0.331350\pi$
$659$	25.9476	1.01077	0.505387	+	0.862893i	$0.331350\pi$
$660$	0	0
$661$	25.8290	1.00463	0.502317	−	0.864684i	$-0.332481\pi$
$661$	25.8290	1.00463	0.502317	+	0.864684i	$0.332481\pi$
$662$	0	0
$663$	6.83519	0.265457
$664$	0	0
$665$	26.0254	1.00922
$666$	0	0
$667$	5.55655	0.215151
$668$	0	0
$669$	−44.3209	−1.71355
$670$	0	0
$671$	39.7850	1.53588
$672$	0	0
$673$	34.5375	1.33132	0.665662	−	0.746254i	$-0.268149\pi$
$673$	34.5375	1.33132	0.665662	+	0.746254i	$0.268149\pi$
$674$	0	0
$675$	−31.6061	−1.21652
$676$	0	0
$677$	24.4793	0.940817	0.470408	−	0.882449i	$-0.344107\pi$
$677$	24.4793	0.940817	0.470408	+	0.882449i	$0.344107\pi$
$678$	0	0
$679$	21.1891	0.813164
$680$	0	0
$681$	−35.4269	−1.35756
$682$	0	0
$683$	−35.2697	−1.34956	−0.674779	−	0.738020i	$-0.735761\pi$
$683$	−35.2697	−1.34956	−0.674779	+	0.738020i	$0.735761\pi$
$684$	0	0
$685$	6.97555	0.266522
$686$	0	0
$687$	82.3704	3.14263
$688$	0	0
$689$	−2.05695	−0.0783635
$690$	0	0
$691$	−11.3110	−0.430292	−0.215146	−	0.976582i	$-0.569023\pi$
$691$	−11.3110	−0.430292	−0.215146	+	0.976582i	$0.569023\pi$
$692$	0	0
$693$	−74.9141	−2.84575
$694$	0	0
$695$	−15.1838	−0.575954
$696$	0	0
$697$	−8.48795	−0.321504
$698$	0	0
$699$	−65.4287	−2.47474
$700$	0	0
$701$	35.1281	1.32677	0.663385	−	0.748278i	$-0.269119\pi$
$701$	35.1281	1.32677	0.663385	+	0.748278i	$0.269119\pi$
$702$	0	0
$703$	5.08797	0.191896
$704$	0	0
$705$	50.1625	1.88923
$706$	0	0
$707$	13.4934	0.507472
$708$	0	0
$709$	31.8815	1.19734	0.598668	−	0.800998i	$-0.295697\pi$
$709$	31.8815	1.19734	0.598668	+	0.800998i	$0.295697\pi$
$710$	0	0
$711$	−5.97812	−0.224197
$712$	0	0
$713$	−64.1916	−2.40400
$714$	0	0
$715$	6.75637	0.252674
$716$	0	0
$717$	14.3509	0.535944
$718$	0	0
$719$	23.0895	0.861094	0.430547	−	0.902568i	$-0.358321\pi$
$719$	23.0895	0.861094	0.430547	+	0.902568i	$0.358321\pi$
$720$	0	0
$721$	−19.1811	−0.714343
$722$	0	0
$723$	−7.58707	−0.282166
$724$	0	0
$725$	−1.65222	−0.0613621
$726$	0	0
$727$	−8.94416	−0.331721	−0.165860	−	0.986149i	$-0.553040\pi$
$727$	−8.94416	−0.331721	−0.165860	+	0.986149i	$0.553040\pi$
$728$	0	0
$729$	83.0680	3.07659
$730$	0	0
$731$	−14.1958	−0.525049
$732$	0	0
$733$	33.0905	1.22223	0.611113	−	0.791543i	$-0.290722\pi$
$733$	33.0905	1.22223	0.611113	+	0.791543i	$0.290722\pi$
$734$	0	0
$735$	−7.74748	−0.285770
$736$	0	0
$737$	15.4993	0.570925
$738$	0	0
$739$	−14.9793	−0.551022	−0.275511	−	0.961298i	$-0.588847\pi$
$739$	−14.9793	−0.551022	−0.275511	+	0.961298i	$0.588847\pi$
$740$	0	0
$741$	−20.1284	−0.739434
$742$	0	0
$743$	6.35093	0.232993	0.116497	−	0.993191i	$-0.462834\pi$
$743$	6.35093	0.232993	0.116497	+	0.993191i	$0.462834\pi$
$744$	0	0
$745$	−8.29100	−0.303759
$746$	0	0
$747$	−8.61179	−0.315089
$748$	0	0
$749$	−20.5898	−0.752336
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−22.4160	−0.816883
$754$	0	0
$755$	9.26598	0.337223
$756$	0	0
$757$	8.17076	0.296971	0.148486	−	0.988915i	$-0.452560\pi$
$757$	8.17076	0.296971	0.148486	+	0.988915i	$0.452560\pi$
$758$	0	0
$759$	−83.5658	−3.03324
$760$	0	0
$761$	49.0768	1.77903	0.889516	−	0.456905i	$-0.151042\pi$
$761$	49.0768	1.77903	0.889516	+	0.456905i	$0.151042\pi$
$762$	0	0
$763$	19.2067	0.695330
$764$	0	0
$765$	−29.6892	−1.07342
$766$	0	0
$767$	5.72889	0.206858
$768$	0	0
$769$	40.4042	1.45701	0.728507	−	0.685038i	$-0.240215\pi$
$769$	40.4042	1.45701	0.728507	+	0.685038i	$0.240215\pi$
$770$	0	0
$771$	−75.1227	−2.70548
$772$	0	0
$773$	54.2891	1.95264	0.976322	−	0.216322i	$-0.0694060\pi$
$773$	54.2891	1.95264	0.976322	+	0.216322i	$0.0694060\pi$
$774$	0	0
$775$	19.0872	0.685632
$776$	0	0
$777$	6.46988	0.232106
$778$	0	0
$779$	24.9954	0.895553
$780$	0	0
$781$	−13.4390	−0.480885
$782$	0	0
$783$	14.3813	0.513945
$784$	0	0
$785$	26.7289	0.953994
$786$	0	0
$787$	−25.6894	−0.915727	−0.457863	−	0.889023i	$-0.651385\pi$
$787$	−25.6894	−0.915727	−0.457863	+	0.889023i	$0.651385\pi$
$788$	0	0
$789$	10.3796	0.369524
$790$	0	0
$791$	23.4713	0.834544
$792$	0	0
$793$	9.88609	0.351065
$794$	0	0
$795$	12.2846	0.435691
$796$	0	0
$797$	−27.0237	−0.957229	−0.478615	−	0.878025i	$-0.658861\pi$
$797$	−27.0237	−0.957229	−0.478615	+	0.878025i	$0.658861\pi$
$798$	0	0
$799$	−18.1364	−0.641620
$800$	0	0
$801$	3.44462	0.121710
$802$	0	0
$803$	27.5230	0.971267
$804$	0	0
$805$	26.8486	0.946290
$806$	0	0
$807$	−55.3209	−1.94739
$808$	0	0
$809$	0.659111	0.0231731	0.0115866	−	0.999933i	$-0.496312\pi$
$809$	0.659111	0.0231731	0.0115866	+	0.999933i	$0.496312\pi$
$810$	0	0
$811$	49.5394	1.73956	0.869781	−	0.493437i	$-0.164260\pi$
$811$	49.5394	1.73956	0.869781	+	0.493437i	$0.164260\pi$
$812$	0	0
$813$	47.8526	1.67826
$814$	0	0
$815$	35.0502	1.22776
$816$	0	0
$817$	41.8039	1.46253
$818$	0	0
$819$	−18.6153	−0.650470
$820$	0	0
$821$	−25.4018	−0.886528	−0.443264	−	0.896391i	$-0.646180\pi$
$821$	−25.4018	−0.886528	−0.443264	+	0.896391i	$0.646180\pi$
$822$	0	0
$823$	−0.895245	−0.0312063	−0.0156031	−	0.999878i	$-0.504967\pi$
$823$	−0.895245	−0.0312063	−0.0156031	+	0.999878i	$0.504967\pi$
$824$	0	0
$825$	24.8480	0.865097
$826$	0	0
$827$	−35.9582	−1.25039	−0.625195	−	0.780468i	$-0.714981\pi$
$827$	−35.9582	−1.25039	−0.625195	+	0.780468i	$0.714981\pi$
$828$	0	0
$829$	32.2894	1.12146	0.560728	−	0.828000i	$-0.310521\pi$
$829$	32.2894	1.12146	0.560728	+	0.828000i	$0.310521\pi$
$830$	0	0
$831$	−34.3459	−1.19145
$832$	0	0
$833$	2.80113	0.0970532
$834$	0	0
$835$	−31.1428	−1.07774
$836$	0	0
$837$	−166.138	−5.74258
$838$	0	0
$839$	0.481597	0.0166266	0.00831330	−	0.999965i	$-0.497354\pi$
$839$	0.481597	0.0166266	0.00831330	+	0.999965i	$0.497354\pi$
$840$	0	0
$841$	−28.2482	−0.974076
$842$	0	0
$843$	−78.0208	−2.68718
$844$	0	0
$845$	−21.1895	−0.728940
$846$	0	0
$847$	10.6142	0.364708
$848$	0	0
$849$	−81.2585	−2.78878
$850$	0	0
$851$	5.24891	0.179930
$852$	0	0
$853$	−14.2214	−0.486932	−0.243466	−	0.969909i	$-0.578284\pi$
$853$	−14.2214	−0.486932	−0.243466	+	0.969909i	$0.578284\pi$
$854$	0	0
$855$	87.4292	2.99002
$856$	0	0
$857$	33.1043	1.13082	0.565411	−	0.824809i	$-0.308718\pi$
$857$	33.1043	1.13082	0.565411	+	0.824809i	$0.308718\pi$
$858$	0	0
$859$	−37.6551	−1.28478	−0.642388	−	0.766380i	$-0.722056\pi$
$859$	−37.6551	−1.28478	−0.642388	+	0.766380i	$0.722056\pi$
$860$	0	0
$861$	31.7843	1.08320
$862$	0	0
$863$	6.60273	0.224759	0.112380	−	0.993665i	$-0.464153\pi$
$863$	6.60273	0.224759	0.112380	+	0.993665i	$0.464153\pi$
$864$	0	0
$865$	−11.0132	−0.374461
$866$	0	0
$867$	−41.6254	−1.41367
$868$	0	0
$869$	2.93759	0.0996508
$870$	0	0
$871$	3.85140	0.130500
$872$	0	0
$873$	71.1824	2.40916
$874$	0	0
$875$	−28.9310	−0.978045
$876$	0	0
$877$	−23.7712	−0.802696	−0.401348	−	0.915926i	$-0.631458\pi$
$877$	−23.7712	−0.802696	−0.401348	+	0.915926i	$0.631458\pi$
$878$	0	0
$879$	46.3313	1.56272
$880$	0	0
$881$	−39.0187	−1.31457	−0.657286	−	0.753641i	$-0.728296\pi$
$881$	−39.0187	−1.31457	−0.657286	+	0.753641i	$0.728296\pi$
$882$	0	0
$883$	−55.6421	−1.87251	−0.936253	−	0.351326i	$-0.885731\pi$
$883$	−55.6421	−1.87251	−0.936253	+	0.351326i	$0.885731\pi$
$884$	0	0
$885$	−34.2145	−1.15011
$886$	0	0
$887$	−24.0861	−0.808731	−0.404366	−	0.914597i	$-0.632508\pi$
$887$	−24.0861	−0.808731	−0.404366	+	0.914597i	$0.632508\pi$
$888$	0	0
$889$	6.55597	0.219880
$890$	0	0
$891$	−121.917	−4.08436
$892$	0	0
$893$	53.4083	1.78724
$894$	0	0
$895$	−44.2711	−1.47982
$896$	0	0
$897$	−20.7651	−0.693326
$898$	0	0
$899$	−8.68496	−0.289660
$900$	0	0
$901$	−4.44155	−0.147970
$902$	0	0
$903$	53.1580	1.76899
$904$	0	0
$905$	1.72874	0.0574653
$906$	0	0
$907$	2.35558	0.0782157	0.0391079	−	0.999235i	$-0.487548\pi$
$907$	2.35558	0.0782157	0.0391079	+	0.999235i	$0.487548\pi$
$908$	0	0
$909$	45.3295	1.50348
$910$	0	0
$911$	45.9853	1.52356	0.761781	−	0.647835i	$-0.224325\pi$
$911$	45.9853	1.52356	0.761781	+	0.647835i	$0.224325\pi$
$912$	0	0
$913$	4.23175	0.140050
$914$	0	0
$915$	−59.0424	−1.95188
$916$	0	0
$917$	7.05748	0.233059
$918$	0	0
$919$	8.45619	0.278944	0.139472	−	0.990226i	$-0.455459\pi$
$919$	8.45619	0.278944	0.139472	+	0.990226i	$0.455459\pi$
$920$	0	0
$921$	−36.7067	−1.20953
$922$	0	0
$923$	−3.33942	−0.109919
$924$	0	0
$925$	−1.56075	−0.0513171
$926$	0	0
$927$	−64.4368	−2.11638
$928$	0	0
$929$	−6.89207	−0.226122	−0.113061	−	0.993588i	$-0.536065\pi$
$929$	−6.89207	−0.226122	−0.113061	+	0.993588i	$0.536065\pi$
$930$	0	0
$931$	−8.24879	−0.270343
$932$	0	0
$933$	35.5682	1.16445
$934$	0	0
$935$	14.5890	0.477111
$936$	0	0
$937$	7.87803	0.257364	0.128682	−	0.991686i	$-0.458925\pi$
$937$	7.87803	0.257364	0.128682	+	0.991686i	$0.458925\pi$
$938$	0	0
$939$	98.8742	3.22664
$940$	0	0
$941$	23.9771	0.781633	0.390816	−	0.920469i	$-0.372193\pi$
$941$	23.9771	0.781633	0.390816	+	0.920469i	$0.372193\pi$
$942$	0	0
$943$	25.7861	0.839711
$944$	0	0
$945$	69.4887	2.26047
$946$	0	0
$947$	24.7491	0.804239	0.402119	−	0.915587i	$-0.368274\pi$
$947$	24.7491	0.804239	0.402119	+	0.915587i	$0.368274\pi$
$948$	0	0
$949$	6.83914	0.222008
$950$	0	0
$951$	−0.292093	−0.00947178
$952$	0	0
$953$	21.4022	0.693285	0.346643	−	0.937997i	$-0.387322\pi$
$953$	21.4022	0.693285	0.346643	+	0.937997i	$0.387322\pi$
$954$	0	0
$955$	28.6940	0.928517
$956$	0	0
$957$	−11.3062	−0.365479
$958$	0	0
$959$	9.44408	0.304965
$960$	0	0
$961$	69.3323	2.23653
$962$	0	0
$963$	−69.1692	−2.22895
$964$	0	0
$965$	−21.4939	−0.691913
$966$	0	0
$967$	−25.4482	−0.818359	−0.409179	−	0.912454i	$-0.634185\pi$
$967$	−25.4482	−0.818359	−0.409179	+	0.912454i	$0.634185\pi$
$968$	0	0
$969$	−43.4630	−1.39623
$970$	0	0
$971$	43.8012	1.40565	0.702823	−	0.711365i	$-0.251923\pi$
$971$	43.8012	1.40565	0.702823	+	0.711365i	$0.251923\pi$
$972$	0	0
$973$	−20.5571	−0.659030
$974$	0	0
$975$	6.17443	0.197740
$976$	0	0
$977$	−4.93318	−0.157826	−0.0789132	−	0.996881i	$-0.525145\pi$
$977$	−4.93318	−0.157826	−0.0789132	+	0.996881i	$0.525145\pi$
$978$	0	0
$979$	−1.69265	−0.0540973
$980$	0	0
$981$	64.5228	2.06005
$982$	0	0
$983$	56.0776	1.78860	0.894299	−	0.447469i	$-0.147675\pi$
$983$	56.0776	1.78860	0.894299	+	0.447469i	$0.147675\pi$
$984$	0	0
$985$	−37.1785	−1.18461
$986$	0	0
$987$	67.9142	2.16173
$988$	0	0
$989$	43.1263	1.37133
$990$	0	0
$991$	31.5887	1.00345	0.501725	−	0.865027i	$-0.332699\pi$
$991$	31.5887	1.00345	0.501725	+	0.865027i	$0.332699\pi$
$992$	0	0
$993$	−91.5178	−2.90423
$994$	0	0
$995$	0.225737	0.00715634
$996$	0	0
$997$	57.7241	1.82814	0.914070	−	0.405557i	$-0.132922\pi$
$997$	57.7241	1.82814	0.914070	+	0.405557i	$0.132922\pi$
$998$	0	0
$999$	13.5850	0.429812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.49	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.49	✓	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)

\(f(q)\)	\(=\)	\(q+3.31674 q^{3} +1.75910 q^{5} +2.38162 q^{7} +8.00078 q^{9} +O(q^{10})\)	\(q+3.31674 q^{3} +1.75910 q^{5} +2.38162 q^{7} +8.00078 q^{9} -3.93150 q^{11} -0.976931 q^{13} +5.83449 q^{15} -2.10948 q^{17} +6.21202 q^{19} +7.89923 q^{21} +6.40852 q^{23} -1.90555 q^{25} +16.5863 q^{27} +0.867057 q^{29} -10.0166 q^{31} -13.0398 q^{33} +4.18952 q^{35} +0.819052 q^{37} -3.24023 q^{39} +4.02372 q^{41} +6.72951 q^{43} +14.0742 q^{45} +8.59758 q^{47} -1.32788 q^{49} -6.99660 q^{51} +2.10552 q^{53} -6.91592 q^{55} +20.6037 q^{57} -5.86417 q^{59} -10.1195 q^{61} +19.0548 q^{63} -1.71852 q^{65} -3.94234 q^{67} +21.2554 q^{69} +3.41828 q^{71} -7.00064 q^{73} -6.32024 q^{75} -9.36335 q^{77} -0.747192 q^{79} +31.0102 q^{81} -1.07637 q^{83} -3.71079 q^{85} +2.87580 q^{87} +0.430535 q^{89} -2.32668 q^{91} -33.2225 q^{93} +10.9276 q^{95} +8.89693 q^{97} -31.4551 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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