LMFDB - Embedded newform 6008.2.a.b.1.16

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:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q-1.52846 q^{3} +0.276340 q^{5} +2.37933 q^{7} -0.663805 q^{9} +O(q^{10})$	$q-1.52846 q^{3} +0.276340 q^{5} +2.37933 q^{7} -0.663805 q^{9} -0.982619 q^{11} -5.12922 q^{13} -0.422376 q^{15} +7.66490 q^{17} +4.67902 q^{19} -3.63671 q^{21} -7.09647 q^{23} -4.92364 q^{25} +5.59999 q^{27} +2.66042 q^{29} -5.74196 q^{31} +1.50189 q^{33} +0.657504 q^{35} -0.273473 q^{37} +7.83981 q^{39} -4.12965 q^{41} +7.07236 q^{43} -0.183436 q^{45} +12.1984 q^{47} -1.33880 q^{49} -11.7155 q^{51} -0.726931 q^{53} -0.271537 q^{55} -7.15171 q^{57} -6.80195 q^{59} +5.98444 q^{61} -1.57941 q^{63} -1.41741 q^{65} -10.6673 q^{67} +10.8467 q^{69} -8.23431 q^{71} +15.4776 q^{73} +7.52559 q^{75} -2.33797 q^{77} -12.1624 q^{79} -6.56795 q^{81} -8.56428 q^{83} +2.11812 q^{85} -4.06634 q^{87} -16.3251 q^{89} -12.2041 q^{91} +8.77637 q^{93} +1.29300 q^{95} +6.63657 q^{97} +0.652267 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * 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
$3$	−1.52846	−0.882458	−0.441229	−	0.897395i	$-0.645457\pi$
$3$	−1.52846	−0.882458	−0.441229	+	0.897395i	$0.645457\pi$
$4$	0	0
$5$	0.276340	0.123583	0.0617916	−	0.998089i	$-0.480319\pi$
$5$	0.276340	0.123583	0.0617916	+	0.998089i	$0.480319\pi$
$6$	0	0
$7$	2.37933	0.899302	0.449651	−	0.893204i	$-0.351548\pi$
$7$	2.37933	0.899302	0.449651	+	0.893204i	$0.351548\pi$
$8$	0	0
$9$	−0.663805	−0.221268
$10$	0	0
$11$	−0.982619	−0.296271	−0.148135	−	0.988967i	$-0.547327\pi$
$11$	−0.982619	−0.296271	−0.148135	+	0.988967i	$0.547327\pi$
$12$	0	0
$13$	−5.12922	−1.42259	−0.711295	−	0.702894i	$-0.751891\pi$
$13$	−5.12922	−1.42259	−0.711295	+	0.702894i	$0.751891\pi$
$14$	0	0
$15$	−0.422376	−0.109057
$16$	0	0
$17$	7.66490	1.85901	0.929505	−	0.368808i	$-0.120234\pi$
$17$	7.66490	1.85901	0.929505	+	0.368808i	$0.120234\pi$
$18$	0	0
$19$	4.67902	1.07344	0.536721	−	0.843760i	$-0.319663\pi$
$19$	4.67902	1.07344	0.536721	+	0.843760i	$0.319663\pi$
$20$	0	0
$21$	−3.63671	−0.793596
$22$	0	0
$23$	−7.09647	−1.47972	−0.739858	−	0.672763i	$-0.765107\pi$
$23$	−7.09647	−1.47972	−0.739858	+	0.672763i	$0.765107\pi$
$24$	0	0
$25$	−4.92364	−0.984727
$26$	0	0
$27$	5.59999	1.07772
$28$	0	0
$29$	2.66042	0.494027	0.247013	−	0.969012i	$-0.420551\pi$
$29$	2.66042	0.494027	0.247013	+	0.969012i	$0.420551\pi$
$30$	0	0
$31$	−5.74196	−1.03129	−0.515644	−	0.856803i	$-0.672447\pi$
$31$	−5.74196	−1.03129	−0.515644	+	0.856803i	$0.672447\pi$
$32$	0	0
$33$	1.50189	0.261446
$34$	0	0
$35$	0.657504	0.111139
$36$	0	0
$37$	−0.273473	−0.0449587	−0.0224793	−	0.999747i	$-0.507156\pi$
$37$	−0.273473	−0.0449587	−0.0224793	+	0.999747i	$0.507156\pi$
$38$	0	0
$39$	7.83981	1.25538
$40$	0	0
$41$	−4.12965	−0.644944	−0.322472	−	0.946579i	$-0.604514\pi$
$41$	−4.12965	−0.644944	−0.322472	+	0.946579i	$0.604514\pi$
$42$	0	0
$43$	7.07236	1.07852	0.539262	−	0.842138i	$-0.318703\pi$
$43$	7.07236	1.07852	0.539262	+	0.842138i	$0.318703\pi$
$44$	0	0
$45$	−0.183436	−0.0273451
$46$	0	0
$47$	12.1984	1.77932	0.889659	−	0.456626i	$-0.150942\pi$
$47$	12.1984	1.77932	0.889659	+	0.456626i	$0.150942\pi$
$48$	0	0
$49$	−1.33880	−0.191257
$50$	0	0
$51$	−11.7155	−1.64050
$52$	0	0
$53$	−0.726931	−0.0998516	−0.0499258	−	0.998753i	$-0.515898\pi$
$53$	−0.726931	−0.0998516	−0.0499258	+	0.998753i	$0.515898\pi$
$54$	0	0
$55$	−0.271537	−0.0366141
$56$	0	0
$57$	−7.15171	−0.947267
$58$	0	0
$59$	−6.80195	−0.885538	−0.442769	−	0.896636i	$-0.646004\pi$
$59$	−6.80195	−0.885538	−0.442769	+	0.896636i	$0.646004\pi$
$60$	0	0
$61$	5.98444	0.766229	0.383114	−	0.923701i	$-0.374851\pi$
$61$	5.98444	0.766229	0.383114	+	0.923701i	$0.374851\pi$
$62$	0	0
$63$	−1.57941	−0.198987
$64$	0	0
$65$	−1.41741	−0.175808
$66$	0	0
$67$	−10.6673	−1.30322	−0.651611	−	0.758553i	$-0.725907\pi$
$67$	−10.6673	−1.30322	−0.651611	+	0.758553i	$0.725907\pi$
$68$	0	0
$69$	10.8467	1.30579
$70$	0	0
$71$	−8.23431	−0.977232	−0.488616	−	0.872499i	$-0.662498\pi$
$71$	−8.23431	−0.977232	−0.488616	+	0.872499i	$0.662498\pi$
$72$	0	0
$73$	15.4776	1.81152	0.905758	−	0.423796i	$-0.139303\pi$
$73$	15.4776	1.81152	0.905758	+	0.423796i	$0.139303\pi$
$74$	0	0
$75$	7.52559	0.868980
$76$	0	0
$77$	−2.33797	−0.266437
$78$	0	0
$79$	−12.1624	−1.36837	−0.684187	−	0.729307i	$-0.739843\pi$
$79$	−12.1624	−1.36837	−0.684187	+	0.729307i	$0.739843\pi$
$80$	0	0
$81$	−6.56795	−0.729772
$82$	0	0
$83$	−8.56428	−0.940052	−0.470026	−	0.882653i	$-0.655755\pi$
$83$	−8.56428	−0.940052	−0.470026	+	0.882653i	$0.655755\pi$
$84$	0	0
$85$	2.11812	0.229742
$86$	0	0
$87$	−4.06634	−0.435958
$88$	0	0
$89$	−16.3251	−1.73046	−0.865230	−	0.501374i	$-0.832828\pi$
$89$	−16.3251	−1.73046	−0.865230	+	0.501374i	$0.832828\pi$
$90$	0	0
$91$	−12.2041	−1.27934
$92$	0	0
$93$	8.77637	0.910067
$94$	0	0
$95$	1.29300	0.132659
$96$	0	0
$97$	6.63657	0.673841	0.336921	−	0.941533i	$-0.390615\pi$
$97$	6.63657	0.673841	0.336921	+	0.941533i	$0.390615\pi$
$98$	0	0
$99$	0.652267	0.0655553
$100$	0	0
$101$	10.3406	1.02893	0.514466	−	0.857511i	$-0.327990\pi$
$101$	10.3406	1.02893	0.514466	+	0.857511i	$0.327990\pi$
$102$	0	0
$103$	2.66191	0.262286	0.131143	−	0.991363i	$-0.458135\pi$
$103$	2.66191	0.262286	0.131143	+	0.991363i	$0.458135\pi$
$104$	0	0
$105$	−1.00497	−0.0980750
$106$	0	0
$107$	−9.29618	−0.898695	−0.449348	−	0.893357i	$-0.648344\pi$
$107$	−9.29618	−0.898695	−0.449348	+	0.893357i	$0.648344\pi$
$108$	0	0
$109$	11.0187	1.05540	0.527698	−	0.849432i	$-0.323055\pi$
$109$	11.0187	1.05540	0.527698	+	0.849432i	$0.323055\pi$
$110$	0	0
$111$	0.417993	0.0396741
$112$	0	0
$113$	1.38536	0.130324	0.0651621	−	0.997875i	$-0.479244\pi$
$113$	1.38536	0.130324	0.0651621	+	0.997875i	$0.479244\pi$
$114$	0	0
$115$	−1.96104	−0.182868
$116$	0	0
$117$	3.40480	0.314774
$118$	0	0
$119$	18.2373	1.67181
$120$	0	0
$121$	−10.0345	−0.912224
$122$	0	0
$123$	6.31202	0.569136
$124$	0	0
$125$	−2.74230	−0.245279
$126$	0	0
$127$	1.69029	0.149989	0.0749945	−	0.997184i	$-0.476106\pi$
$127$	1.69029	0.149989	0.0749945	+	0.997184i	$0.476106\pi$
$128$	0	0
$129$	−10.8098	−0.951752
$130$	0	0
$131$	11.3218	0.989186	0.494593	−	0.869125i	$-0.335317\pi$
$131$	11.3218	0.989186	0.494593	+	0.869125i	$0.335317\pi$
$132$	0	0
$133$	11.1329	0.965347
$134$	0	0
$135$	1.54750	0.133188
$136$	0	0
$137$	−8.67204	−0.740902	−0.370451	−	0.928852i	$-0.620797\pi$
$137$	−8.67204	−0.740902	−0.370451	+	0.928852i	$0.620797\pi$
$138$	0	0
$139$	14.6663	1.24398	0.621988	−	0.783027i	$-0.286325\pi$
$139$	14.6663	1.24398	0.621988	+	0.783027i	$0.286325\pi$
$140$	0	0
$141$	−18.6448	−1.57017
$142$	0	0
$143$	5.04007	0.421472
$144$	0	0
$145$	0.735180	0.0610534
$146$	0	0
$147$	2.04630	0.168776
$148$	0	0
$149$	−11.9171	−0.976291	−0.488145	−	0.872762i	$-0.662326\pi$
$149$	−11.9171	−0.976291	−0.488145	+	0.872762i	$0.662326\pi$
$150$	0	0
$151$	−7.77060	−0.632362	−0.316181	−	0.948699i	$-0.602401\pi$
$151$	−7.77060	−0.632362	−0.316181	+	0.948699i	$0.602401\pi$
$152$	0	0
$153$	−5.08800	−0.411340
$154$	0	0
$155$	−1.58674	−0.127450
$156$	0	0
$157$	−14.9081	−1.18980	−0.594900	−	0.803800i	$-0.702808\pi$
$157$	−14.9081	−1.18980	−0.594900	+	0.803800i	$0.702808\pi$
$158$	0	0
$159$	1.11109	0.0881148
$160$	0	0
$161$	−16.8848	−1.33071
$162$	0	0
$163$	−16.2000	−1.26889	−0.634443	−	0.772970i	$-0.718770\pi$
$163$	−16.2000	−1.26889	−0.634443	+	0.772970i	$0.718770\pi$
$164$	0	0
$165$	0.415034	0.0323104
$166$	0	0
$167$	0.0637168	0.00493055	0.00246528	−	0.999997i	$-0.499215\pi$
$167$	0.0637168	0.00493055	0.00246528	+	0.999997i	$0.499215\pi$
$168$	0	0
$169$	13.3089	1.02376
$170$	0	0
$171$	−3.10596	−0.237519
$172$	0	0
$173$	23.8595	1.81401	0.907003	−	0.421124i	$-0.138364\pi$
$173$	23.8595	1.81401	0.907003	+	0.421124i	$0.138364\pi$
$174$	0	0
$175$	−11.7149	−0.885567
$176$	0	0
$177$	10.3965	0.781450
$178$	0	0
$179$	−14.2960	−1.06853	−0.534267	−	0.845316i	$-0.679412\pi$
$179$	−14.2960	−1.06853	−0.534267	+	0.845316i	$0.679412\pi$
$180$	0	0
$181$	16.7621	1.24592	0.622960	−	0.782254i	$-0.285930\pi$
$181$	16.7621	1.24592	0.622960	+	0.782254i	$0.285930\pi$
$182$	0	0
$183$	−9.14698	−0.676165
$184$	0	0
$185$	−0.0755716	−0.00555614
$186$	0	0
$187$	−7.53167	−0.550770
$188$	0	0
$189$	13.3242	0.969193
$190$	0	0
$191$	−14.9060	−1.07856	−0.539279	−	0.842127i	$-0.681303\pi$
$191$	−14.9060	−1.07856	−0.539279	+	0.842127i	$0.681303\pi$
$192$	0	0
$193$	−4.51590	−0.325062	−0.162531	−	0.986703i	$-0.551966\pi$
$193$	−4.51590	−0.325062	−0.162531	+	0.986703i	$0.551966\pi$
$194$	0	0
$195$	2.16646	0.155143
$196$	0	0
$197$	17.9542	1.27919	0.639593	−	0.768714i	$-0.279103\pi$
$197$	17.9542	1.27919	0.639593	+	0.768714i	$0.279103\pi$
$198$	0	0
$199$	12.5885	0.892378	0.446189	−	0.894939i	$-0.352781\pi$
$199$	12.5885	0.892378	0.446189	+	0.894939i	$0.352781\pi$
$200$	0	0
$201$	16.3046	1.15004
$202$	0	0
$203$	6.33000	0.444279
$204$	0	0
$205$	−1.14119	−0.0797042
$206$	0	0
$207$	4.71068	0.327415
$208$	0	0
$209$	−4.59769	−0.318029
$210$	0	0
$211$	−1.10396	−0.0760001	−0.0380000	−	0.999278i	$-0.512099\pi$
$211$	−1.10396	−0.0760001	−0.0380000	+	0.999278i	$0.512099\pi$
$212$	0	0
$213$	12.5858	0.862366
$214$	0	0
$215$	1.95438	0.133287
$216$	0	0
$217$	−13.6620	−0.927438
$218$	0	0
$219$	−23.6569	−1.59859
$220$	0	0
$221$	−39.3149	−2.64461
$222$	0	0
$223$	−27.8136	−1.86254	−0.931268	−	0.364335i	$-0.881296\pi$
$223$	−27.8136	−1.86254	−0.931268	+	0.364335i	$0.881296\pi$
$224$	0	0
$225$	3.26834	0.217889
$226$	0	0
$227$	−15.4401	−1.02479	−0.512397	−	0.858749i	$-0.671242\pi$
$227$	−15.4401	−1.02479	−0.512397	+	0.858749i	$0.671242\pi$
$228$	0	0
$229$	9.73578	0.643359	0.321679	−	0.946849i	$-0.395753\pi$
$229$	9.73578	0.643359	0.321679	+	0.946849i	$0.395753\pi$
$230$	0	0
$231$	3.57350	0.235119
$232$	0	0
$233$	−10.8528	−0.710988	−0.355494	−	0.934679i	$-0.615687\pi$
$233$	−10.8528	−0.710988	−0.355494	+	0.934679i	$0.615687\pi$
$234$	0	0
$235$	3.37091	0.219894
$236$	0	0
$237$	18.5897	1.20753
$238$	0	0
$239$	−25.7278	−1.66419	−0.832095	−	0.554632i	$-0.812859\pi$
$239$	−25.7278	−1.66419	−0.832095	+	0.554632i	$0.812859\pi$
$240$	0	0
$241$	−8.19705	−0.528018	−0.264009	−	0.964520i	$-0.585045\pi$
$241$	−8.19705	−0.528018	−0.264009	+	0.964520i	$0.585045\pi$
$242$	0	0
$243$	−6.76110	−0.433725
$244$	0	0
$245$	−0.369964	−0.0236361
$246$	0	0
$247$	−23.9997	−1.52707
$248$	0	0
$249$	13.0902	0.829556
$250$	0	0
$251$	1.07869	0.0680862	0.0340431	−	0.999420i	$-0.489162\pi$
$251$	1.07869	0.0680862	0.0340431	+	0.999420i	$0.489162\pi$
$252$	0	0
$253$	6.97312	0.438397
$254$	0	0
$255$	−3.23747	−0.202738
$256$	0	0
$257$	−25.7180	−1.60425	−0.802124	−	0.597158i	$-0.796297\pi$
$257$	−25.7180	−1.60425	−0.802124	+	0.597158i	$0.796297\pi$
$258$	0	0
$259$	−0.650682	−0.0404314
$260$	0	0
$261$	−1.76600	−0.109313
$262$	0	0
$263$	27.0811	1.66990	0.834948	−	0.550330i	$-0.185498\pi$
$263$	27.0811	1.66990	0.834948	+	0.550330i	$0.185498\pi$
$264$	0	0
$265$	−0.200880	−0.0123400
$266$	0	0
$267$	24.9523	1.52706
$268$	0	0
$269$	18.3997	1.12185	0.560923	−	0.827868i	$-0.310446\pi$
$269$	18.3997	1.12185	0.560923	+	0.827868i	$0.310446\pi$
$270$	0	0
$271$	−17.9111	−1.08802	−0.544010	−	0.839079i	$-0.683095\pi$
$271$	−17.9111	−1.08802	−0.544010	+	0.839079i	$0.683095\pi$
$272$	0	0
$273$	18.6535	1.12896
$274$	0	0
$275$	4.83806	0.291746
$276$	0	0
$277$	−7.11279	−0.427366	−0.213683	−	0.976903i	$-0.568546\pi$
$277$	−7.11279	−0.427366	−0.213683	+	0.976903i	$0.568546\pi$
$278$	0	0
$279$	3.81155	0.228191
$280$	0	0
$281$	−15.9208	−0.949758	−0.474879	−	0.880051i	$-0.657508\pi$
$281$	−15.9208	−0.949758	−0.474879	+	0.880051i	$0.657508\pi$
$282$	0	0
$283$	−15.9716	−0.949415	−0.474708	−	0.880144i	$-0.657446\pi$
$283$	−15.9716	−0.949415	−0.474708	+	0.880144i	$0.657446\pi$
$284$	0	0
$285$	−1.97630	−0.117066
$286$	0	0
$287$	−9.82580	−0.579999
$288$	0	0
$289$	41.7507	2.45592
$290$	0	0
$291$	−10.1437	−0.594636
$292$	0	0
$293$	−12.1943	−0.712399	−0.356200	−	0.934410i	$-0.615928\pi$
$293$	−12.1943	−0.712399	−0.356200	+	0.934410i	$0.615928\pi$
$294$	0	0
$295$	−1.87965	−0.109438
$296$	0	0
$297$	−5.50265	−0.319296
$298$	0	0
$299$	36.3994	2.10503
$300$	0	0
$301$	16.8275	0.969919
$302$	0	0
$303$	−15.8053	−0.907988
$304$	0	0
$305$	1.65374	0.0946930
$306$	0	0
$307$	−25.9628	−1.48178	−0.740889	−	0.671628i	$-0.765595\pi$
$307$	−25.9628	−1.48178	−0.740889	+	0.671628i	$0.765595\pi$
$308$	0	0
$309$	−4.06863	−0.231456
$310$	0	0
$311$	−9.92858	−0.562998	−0.281499	−	0.959562i	$-0.590832\pi$
$311$	−9.92858	−0.562998	−0.281499	+	0.959562i	$0.590832\pi$
$312$	0	0
$313$	22.4924	1.27135	0.635673	−	0.771958i	$-0.280723\pi$
$313$	22.4924	1.27135	0.635673	+	0.771958i	$0.280723\pi$
$314$	0	0
$315$	−0.436455	−0.0245914
$316$	0	0
$317$	−6.16889	−0.346480	−0.173240	−	0.984880i	$-0.555424\pi$
$317$	−6.16889	−0.346480	−0.173240	+	0.984880i	$0.555424\pi$
$318$	0	0
$319$	−2.61417	−0.146366
$320$	0	0
$321$	14.2088	0.793061
$322$	0	0
$323$	35.8642	1.99554
$324$	0	0
$325$	25.2544	1.40086
$326$	0	0
$327$	−16.8416	−0.931343
$328$	0	0
$329$	29.0240	1.60014
$330$	0	0
$331$	−24.3642	−1.33918	−0.669588	−	0.742733i	$-0.733529\pi$
$331$	−24.3642	−1.33918	−0.669588	+	0.742733i	$0.733529\pi$
$332$	0	0
$333$	0.181533	0.00994794
$334$	0	0
$335$	−2.94782	−0.161056
$336$	0	0
$337$	−27.5839	−1.50259	−0.751294	−	0.659967i	$-0.770570\pi$
$337$	−27.5839	−1.50259	−0.751294	+	0.659967i	$0.770570\pi$
$338$	0	0
$339$	−2.11748	−0.115006
$340$	0	0
$341$	5.64216	0.305540
$342$	0	0
$343$	−19.8407	−1.07130
$344$	0	0
$345$	2.99738	0.161373
$346$	0	0
$347$	16.1313	0.865972	0.432986	−	0.901401i	$-0.357460\pi$
$347$	16.1313	0.865972	0.432986	+	0.901401i	$0.357460\pi$
$348$	0	0
$349$	−10.0145	−0.536064	−0.268032	−	0.963410i	$-0.586373\pi$
$349$	−10.0145	−0.536064	−0.268032	+	0.963410i	$0.586373\pi$
$350$	0	0
$351$	−28.7236	−1.53315
$352$	0	0
$353$	18.2704	0.972437	0.486218	−	0.873837i	$-0.338376\pi$
$353$	18.2704	0.972437	0.486218	+	0.873837i	$0.338376\pi$
$354$	0	0
$355$	−2.27547	−0.120769
$356$	0	0
$357$	−27.8750	−1.47530
$358$	0	0
$359$	36.9280	1.94898	0.974492	−	0.224421i	$-0.0720491\pi$
$359$	36.9280	1.94898	0.974492	+	0.224421i	$0.0720491\pi$
$360$	0	0
$361$	2.89325	0.152276
$362$	0	0
$363$	15.3373	0.804999
$364$	0	0
$365$	4.27708	0.223873
$366$	0	0
$367$	−13.0694	−0.682219	−0.341109	−	0.940024i	$-0.610803\pi$
$367$	−13.0694	−0.682219	−0.341109	+	0.940024i	$0.610803\pi$
$368$	0	0
$369$	2.74129	0.142706
$370$	0	0
$371$	−1.72961	−0.0897967
$372$	0	0
$373$	22.5438	1.16728	0.583638	−	0.812014i	$-0.301629\pi$
$373$	22.5438	1.16728	0.583638	+	0.812014i	$0.301629\pi$
$374$	0	0
$375$	4.19150	0.216448
$376$	0	0
$377$	−13.6459	−0.702797
$378$	0	0
$379$	8.21538	0.421996	0.210998	−	0.977487i	$-0.432329\pi$
$379$	8.21538	0.421996	0.210998	+	0.977487i	$0.432329\pi$
$380$	0	0
$381$	−2.58354	−0.132359
$382$	0	0
$383$	18.7975	0.960508	0.480254	−	0.877129i	$-0.340544\pi$
$383$	18.7975	0.960508	0.480254	+	0.877129i	$0.340544\pi$
$384$	0	0
$385$	−0.646076	−0.0329271
$386$	0	0
$387$	−4.69467	−0.238643
$388$	0	0
$389$	6.78801	0.344166	0.172083	−	0.985082i	$-0.444950\pi$
$389$	6.78801	0.344166	0.172083	+	0.985082i	$0.444950\pi$
$390$	0	0
$391$	−54.3937	−2.75081
$392$	0	0
$393$	−17.3049	−0.872915
$394$	0	0
$395$	−3.36095	−0.169108
$396$	0	0
$397$	−6.42103	−0.322262	−0.161131	−	0.986933i	$-0.551514\pi$
$397$	−6.42103	−0.322262	−0.161131	+	0.986933i	$0.551514\pi$
$398$	0	0
$399$	−17.0163	−0.851878
$400$	0	0
$401$	−5.42818	−0.271070	−0.135535	−	0.990773i	$-0.543275\pi$
$401$	−5.42818	−0.271070	−0.135535	+	0.990773i	$0.543275\pi$
$402$	0	0
$403$	29.4518	1.46710
$404$	0	0
$405$	−1.81499	−0.0901875
$406$	0	0
$407$	0.268720	0.0133199
$408$	0	0
$409$	−31.8163	−1.57321	−0.786607	−	0.617453i	$-0.788164\pi$
$409$	−31.8163	−1.57321	−0.786607	+	0.617453i	$0.788164\pi$
$410$	0	0
$411$	13.2549	0.653815
$412$	0	0
$413$	−16.1841	−0.796366
$414$	0	0
$415$	−2.36666	−0.116175
$416$	0	0
$417$	−22.4168	−1.09776
$418$	0	0
$419$	−24.8898	−1.21594	−0.607972	−	0.793958i	$-0.708017\pi$
$419$	−24.8898	−1.21594	−0.607972	+	0.793958i	$0.708017\pi$
$420$	0	0
$421$	−22.1147	−1.07780	−0.538902	−	0.842368i	$-0.681161\pi$
$421$	−22.1147	−1.07780	−0.538902	+	0.842368i	$0.681161\pi$
$422$	0	0
$423$	−8.09736	−0.393707
$424$	0	0
$425$	−37.7392	−1.83062
$426$	0	0
$427$	14.2389	0.689071
$428$	0	0
$429$	−7.70355	−0.371931
$430$	0	0
$431$	−30.2171	−1.45551	−0.727754	−	0.685839i	$-0.759436\pi$
$431$	−30.2171	−1.45551	−0.727754	+	0.685839i	$0.759436\pi$
$432$	0	0
$433$	6.28105	0.301848	0.150924	−	0.988545i	$-0.451775\pi$
$433$	6.28105	0.301848	0.150924	+	0.988545i	$0.451775\pi$
$434$	0	0
$435$	−1.12369	−0.0538770
$436$	0	0
$437$	−33.2045	−1.58839
$438$	0	0
$439$	−37.6291	−1.79594	−0.897969	−	0.440058i	$-0.854958\pi$
$439$	−37.6291	−1.79594	−0.897969	+	0.440058i	$0.854958\pi$
$440$	0	0
$441$	0.888701	0.0423191
$442$	0	0
$443$	−23.5280	−1.11785	−0.558925	−	0.829219i	$-0.688786\pi$
$443$	−23.5280	−1.11785	−0.558925	+	0.829219i	$0.688786\pi$
$444$	0	0
$445$	−4.51129	−0.213856
$446$	0	0
$447$	18.2149	0.861535
$448$	0	0
$449$	6.42606	0.303264	0.151632	−	0.988437i	$-0.451547\pi$
$449$	6.42606	0.303264	0.151632	+	0.988437i	$0.451547\pi$
$450$	0	0
$451$	4.05787	0.191078
$452$	0	0
$453$	11.8771	0.558033
$454$	0	0
$455$	−3.37248	−0.158104
$456$	0	0
$457$	−34.7818	−1.62702	−0.813511	−	0.581549i	$-0.802447\pi$
$457$	−34.7818	−1.62702	−0.813511	+	0.581549i	$0.802447\pi$
$458$	0	0
$459$	42.9233	2.00349
$460$	0	0
$461$	16.7636	0.780761	0.390380	−	0.920654i	$-0.372343\pi$
$461$	16.7636	0.780761	0.390380	+	0.920654i	$0.372343\pi$
$462$	0	0
$463$	−17.1355	−0.796356	−0.398178	−	0.917308i	$-0.630357\pi$
$463$	−17.1355	−0.796356	−0.398178	+	0.917308i	$0.630357\pi$
$464$	0	0
$465$	2.42527	0.112469
$466$	0	0
$467$	−3.92161	−0.181470	−0.0907351	−	0.995875i	$-0.528922\pi$
$467$	−3.92161	−0.181470	−0.0907351	+	0.995875i	$0.528922\pi$
$468$	0	0
$469$	−25.3811	−1.17199
$470$	0	0
$471$	22.7865	1.04995
$472$	0	0
$473$	−6.94943	−0.319535
$474$	0	0
$475$	−23.0378	−1.05705
$476$	0	0
$477$	0.482540	0.0220940
$478$	0	0
$479$	−12.9032	−0.589560	−0.294780	−	0.955565i	$-0.595246\pi$
$479$	−12.9032	−0.589560	−0.294780	+	0.955565i	$0.595246\pi$
$480$	0	0
$481$	1.40270	0.0639578
$482$	0	0
$483$	25.8078	1.17430
$484$	0	0
$485$	1.83395	0.0832754
$486$	0	0
$487$	−31.9698	−1.44869	−0.724345	−	0.689438i	$-0.757858\pi$
$487$	−31.9698	−1.44869	−0.724345	+	0.689438i	$0.757858\pi$
$488$	0	0
$489$	24.7611	1.11974
$490$	0	0
$491$	−18.1637	−0.819716	−0.409858	−	0.912149i	$-0.634422\pi$
$491$	−18.1637	−0.819716	−0.409858	+	0.912149i	$0.634422\pi$
$492$	0	0
$493$	20.3918	0.918401
$494$	0	0
$495$	0.180248	0.00810154
$496$	0	0
$497$	−19.5921	−0.878827
$498$	0	0
$499$	−23.9820	−1.07358	−0.536791	−	0.843715i	$-0.680363\pi$
$499$	−23.9820	−1.07358	−0.536791	+	0.843715i	$0.680363\pi$
$500$	0	0
$501$	−0.0973887	−0.00435100
$502$	0	0
$503$	−29.0278	−1.29429	−0.647143	−	0.762368i	$-0.724037\pi$
$503$	−29.0278	−1.29429	−0.647143	+	0.762368i	$0.724037\pi$
$504$	0	0
$505$	2.85753	0.127159
$506$	0	0
$507$	−20.3421	−0.903426
$508$	0	0
$509$	3.24316	0.143750	0.0718752	−	0.997414i	$-0.477102\pi$
$509$	3.24316	0.143750	0.0718752	+	0.997414i	$0.477102\pi$
$510$	0	0
$511$	36.8263	1.62910
$512$	0	0
$513$	26.2025	1.15687
$514$	0	0
$515$	0.735593	0.0324141
$516$	0	0
$517$	−11.9864	−0.527160
$518$	0	0
$519$	−36.4684	−1.60078
$520$	0	0
$521$	5.00903	0.219450	0.109725	−	0.993962i	$-0.465003\pi$
$521$	5.00903	0.219450	0.109725	+	0.993962i	$0.465003\pi$
$522$	0	0
$523$	10.7377	0.469525	0.234763	−	0.972053i	$-0.424569\pi$
$523$	10.7377	0.469525	0.234763	+	0.972053i	$0.424569\pi$
$524$	0	0
$525$	17.9058	0.781475
$526$	0	0
$527$	−44.0116	−1.91717
$528$	0	0
$529$	27.3599	1.18956
$530$	0	0
$531$	4.51517	0.195942
$532$	0	0
$533$	21.1819	0.917490
$534$	0	0
$535$	−2.56891	−0.111064
$536$	0	0
$537$	21.8509	0.942936
$538$	0	0
$539$	1.31553	0.0566638
$540$	0	0
$541$	−16.9617	−0.729239	−0.364619	−	0.931157i	$-0.618801\pi$
$541$	−16.9617	−0.729239	−0.364619	+	0.931157i	$0.618801\pi$
$542$	0	0
$543$	−25.6203	−1.09947
$544$	0	0
$545$	3.04490	0.130429
$546$	0	0
$547$	−21.5194	−0.920105	−0.460052	−	0.887892i	$-0.652169\pi$
$547$	−21.5194	−0.920105	−0.460052	+	0.887892i	$0.652169\pi$
$548$	0	0
$549$	−3.97250	−0.169542
$550$	0	0
$551$	12.4481	0.530309
$552$	0	0
$553$	−28.9383	−1.23058
$554$	0	0
$555$	0.115508	0.00490306
$556$	0	0
$557$	13.5868	0.575689	0.287845	−	0.957677i	$-0.407061\pi$
$557$	13.5868	0.575689	0.287845	+	0.957677i	$0.407061\pi$
$558$	0	0
$559$	−36.2757	−1.53430
$560$	0	0
$561$	11.5119	0.486032
$562$	0	0
$563$	14.9004	0.627978	0.313989	−	0.949427i	$-0.398335\pi$
$563$	14.9004	0.627978	0.313989	+	0.949427i	$0.398335\pi$
$564$	0	0
$565$	0.382832	0.0161059
$566$	0	0
$567$	−15.6273	−0.656285
$568$	0	0
$569$	36.0502	1.51130	0.755651	−	0.654974i	$-0.227320\pi$
$569$	36.0502	1.51130	0.755651	+	0.654974i	$0.227320\pi$
$570$	0	0
$571$	40.6936	1.70298	0.851488	−	0.524375i	$-0.175701\pi$
$571$	40.6936	1.70298	0.851488	+	0.524375i	$0.175701\pi$
$572$	0	0
$573$	22.7832	0.951782
$574$	0	0
$575$	34.9404	1.45712
$576$	0	0
$577$	−42.6151	−1.77409	−0.887046	−	0.461681i	$-0.847246\pi$
$577$	−42.6151	−1.77409	−0.887046	+	0.461681i	$0.847246\pi$
$578$	0	0
$579$	6.90238	0.286853
$580$	0	0
$581$	−20.3772	−0.845390
$582$	0	0
$583$	0.714295	0.0295831
$584$	0	0
$585$	0.940884	0.0389008
$586$	0	0
$587$	25.7650	1.06343	0.531717	−	0.846922i	$-0.321547\pi$
$587$	25.7650	1.06343	0.531717	+	0.846922i	$0.321547\pi$
$588$	0	0
$589$	−26.8668	−1.10703
$590$	0	0
$591$	−27.4424	−1.12883
$592$	0	0
$593$	−22.5745	−0.927025	−0.463513	−	0.886090i	$-0.653411\pi$
$593$	−22.5745	−0.927025	−0.463513	+	0.886090i	$0.653411\pi$
$594$	0	0
$595$	5.03970	0.206608
$596$	0	0
$597$	−19.2411	−0.787486
$598$	0	0
$599$	8.92050	0.364482	0.182241	−	0.983254i	$-0.441665\pi$
$599$	8.92050	0.364482	0.182241	+	0.983254i	$0.441665\pi$
$600$	0	0
$601$	14.7007	0.599656	0.299828	−	0.953993i	$-0.403071\pi$
$601$	14.7007	0.599656	0.299828	+	0.953993i	$0.403071\pi$
$602$	0	0
$603$	7.08104	0.288362
$604$	0	0
$605$	−2.77293	−0.112735
$606$	0	0
$607$	−5.61731	−0.227999	−0.114000	−	0.993481i	$-0.536366\pi$
$607$	−5.61731	−0.227999	−0.114000	+	0.993481i	$0.536366\pi$
$608$	0	0
$609$	−9.67516	−0.392057
$610$	0	0
$611$	−62.5682	−2.53124
$612$	0	0
$613$	−33.0260	−1.33391	−0.666953	−	0.745100i	$-0.732402\pi$
$613$	−33.0260	−1.33391	−0.666953	+	0.745100i	$0.732402\pi$
$614$	0	0
$615$	1.74427	0.0703356
$616$	0	0
$617$	38.6964	1.55786	0.778930	−	0.627111i	$-0.215763\pi$
$617$	38.6964	1.55786	0.778930	+	0.627111i	$0.215763\pi$
$618$	0	0
$619$	1.27152	0.0511065	0.0255533	−	0.999673i	$-0.491865\pi$
$619$	1.27152	0.0511065	0.0255533	+	0.999673i	$0.491865\pi$
$620$	0	0
$621$	−39.7401	−1.59472
$622$	0	0
$623$	−38.8429	−1.55621
$624$	0	0
$625$	23.8604	0.954415
$626$	0	0
$627$	7.02740	0.280647
$628$	0	0
$629$	−2.09614	−0.0835787
$630$	0	0
$631$	−17.6529	−0.702750	−0.351375	−	0.936235i	$-0.614286\pi$
$631$	−17.6529	−0.702750	−0.351375	+	0.936235i	$0.614286\pi$
$632$	0	0
$633$	1.68737	0.0670668
$634$	0	0
$635$	0.467095	0.0185361
$636$	0	0
$637$	6.86699	0.272080
$638$	0	0
$639$	5.46598	0.216231
$640$	0	0
$641$	−24.0368	−0.949395	−0.474697	−	0.880149i	$-0.657442\pi$
$641$	−24.0368	−0.949395	−0.474697	+	0.880149i	$0.657442\pi$
$642$	0	0
$643$	2.27807	0.0898382	0.0449191	−	0.998991i	$-0.485697\pi$
$643$	2.27807	0.0898382	0.0449191	+	0.998991i	$0.485697\pi$
$644$	0	0
$645$	−2.98719	−0.117621
$646$	0	0
$647$	14.6344	0.575337	0.287668	−	0.957730i	$-0.407120\pi$
$647$	14.6344	0.575337	0.287668	+	0.957730i	$0.407120\pi$
$648$	0	0
$649$	6.68372	0.262359
$650$	0	0
$651$	20.8819	0.818425
$652$	0	0
$653$	−45.3939	−1.77640	−0.888200	−	0.459456i	$-0.848044\pi$
$653$	−45.3939	−1.77640	−0.888200	+	0.459456i	$0.848044\pi$
$654$	0	0
$655$	3.12866	0.122247
$656$	0	0
$657$	−10.2741	−0.400831
$658$	0	0
$659$	16.7090	0.650891	0.325446	−	0.945561i	$-0.394486\pi$
$659$	16.7090	0.650891	0.325446	+	0.945561i	$0.394486\pi$
$660$	0	0
$661$	−2.30234	−0.0895505	−0.0447752	−	0.998997i	$-0.514257\pi$
$661$	−2.30234	−0.0895505	−0.0447752	+	0.998997i	$0.514257\pi$
$662$	0	0
$663$	60.0914	2.33376
$664$	0	0
$665$	3.07648	0.119301
$666$	0	0
$667$	−18.8796	−0.731020
$668$	0	0
$669$	42.5120	1.64361
$670$	0	0
$671$	−5.88042	−0.227011
$672$	0	0
$673$	16.7779	0.646739	0.323369	−	0.946273i	$-0.395184\pi$
$673$	16.7779	0.646739	0.323369	+	0.946273i	$0.395184\pi$
$674$	0	0
$675$	−27.5723	−1.06126
$676$	0	0
$677$	30.0279	1.15407	0.577033	−	0.816721i	$-0.304210\pi$
$677$	30.0279	1.15407	0.577033	+	0.816721i	$0.304210\pi$
$678$	0	0
$679$	15.7906	0.605986
$680$	0	0
$681$	23.5996	0.904337
$682$	0	0
$683$	32.0981	1.22820	0.614099	−	0.789229i	$-0.289519\pi$
$683$	32.0981	1.22820	0.614099	+	0.789229i	$0.289519\pi$
$684$	0	0
$685$	−2.39643	−0.0915631
$686$	0	0
$687$	−14.8808	−0.567737
$688$	0	0
$689$	3.72859	0.142048
$690$	0	0
$691$	−19.4004	−0.738027	−0.369014	−	0.929424i	$-0.620304\pi$
$691$	−19.4004	−0.738027	−0.369014	+	0.929424i	$0.620304\pi$
$692$	0	0
$693$	1.55196	0.0589540
$694$	0	0
$695$	4.05288	0.153734
$696$	0	0
$697$	−31.6534	−1.19896
$698$	0	0
$699$	16.5880	0.627417
$700$	0	0
$701$	−16.9642	−0.640730	−0.320365	−	0.947294i	$-0.603806\pi$
$701$	−16.9642	−0.640730	−0.320365	+	0.947294i	$0.603806\pi$
$702$	0	0
$703$	−1.27959	−0.0482605
$704$	0	0
$705$	−5.15230	−0.194047
$706$	0	0
$707$	24.6038	0.925319
$708$	0	0
$709$	13.7779	0.517440	0.258720	−	0.965952i	$-0.416699\pi$
$709$	13.7779	0.517440	0.258720	+	0.965952i	$0.416699\pi$
$710$	0	0
$711$	8.07345	0.302778
$712$	0	0
$713$	40.7477	1.52601
$714$	0	0
$715$	1.39277	0.0520868
$716$	0	0
$717$	39.3239	1.46858
$718$	0	0
$719$	33.1958	1.23800	0.618998	−	0.785393i	$-0.287539\pi$
$719$	33.1958	1.23800	0.618998	+	0.785393i	$0.287539\pi$
$720$	0	0
$721$	6.33356	0.235874
$722$	0	0
$723$	12.5289	0.465954
$724$	0	0
$725$	−13.0989	−0.486482
$726$	0	0
$727$	12.4300	0.461004	0.230502	−	0.973072i	$-0.425963\pi$
$727$	12.4300	0.461004	0.230502	+	0.973072i	$0.425963\pi$
$728$	0	0
$729$	30.0379	1.11252
$730$	0	0
$731$	54.2089	2.00499
$732$	0	0
$733$	10.0371	0.370727	0.185363	−	0.982670i	$-0.440654\pi$
$733$	10.0371	0.370727	0.185363	+	0.982670i	$0.440654\pi$
$734$	0	0
$735$	0.565475	0.0208579
$736$	0	0
$737$	10.4819	0.386107
$738$	0	0
$739$	39.0083	1.43494	0.717471	−	0.696588i	$-0.245299\pi$
$739$	39.0083	1.43494	0.717471	+	0.696588i	$0.245299\pi$
$740$	0	0
$741$	36.6827	1.34757
$742$	0	0
$743$	−27.3362	−1.00287	−0.501433	−	0.865196i	$-0.667194\pi$
$743$	−27.3362	−1.00287	−0.501433	+	0.865196i	$0.667194\pi$
$744$	0	0
$745$	−3.29319	−0.120653
$746$	0	0
$747$	5.68502	0.208004
$748$	0	0
$749$	−22.1187	−0.808198
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−1.64873	−0.0600832
$754$	0	0
$755$	−2.14733	−0.0781493
$756$	0	0
$757$	23.9456	0.870317	0.435159	−	0.900354i	$-0.356692\pi$
$757$	23.9456	0.870317	0.435159	+	0.900354i	$0.356692\pi$
$758$	0	0
$759$	−10.6582	−0.386866
$760$	0	0
$761$	32.7674	1.18782	0.593909	−	0.804533i	$-0.297584\pi$
$761$	32.7674	1.18782	0.593909	+	0.804533i	$0.297584\pi$
$762$	0	0
$763$	26.2170	0.949120
$764$	0	0
$765$	−1.40602	−0.0508347
$766$	0	0
$767$	34.8887	1.25976
$768$	0	0
$769$	−9.43022	−0.340063	−0.170031	−	0.985439i	$-0.554387\pi$
$769$	−9.43022	−0.340063	−0.170031	+	0.985439i	$0.554387\pi$
$770$	0	0
$771$	39.3090	1.41568
$772$	0	0
$773$	−23.8115	−0.856442	−0.428221	−	0.903674i	$-0.640859\pi$
$773$	−23.8115	−0.856442	−0.428221	+	0.903674i	$0.640859\pi$
$774$	0	0
$775$	28.2713	1.01554
$776$	0	0
$777$	0.994542	0.0356790
$778$	0	0
$779$	−19.3227	−0.692309
$780$	0	0
$781$	8.09118	0.289525
$782$	0	0
$783$	14.8983	0.532421
$784$	0	0
$785$	−4.11972	−0.147039
$786$	0	0
$787$	−23.1542	−0.825359	−0.412679	−	0.910876i	$-0.635407\pi$
$787$	−23.1542	−0.825359	−0.412679	+	0.910876i	$0.635407\pi$
$788$	0	0
$789$	−41.3925	−1.47361
$790$	0	0
$791$	3.29624	0.117201
$792$	0	0
$793$	−30.6955	−1.09003
$794$	0	0
$795$	0.307038	0.0108895
$796$	0	0
$797$	1.66856	0.0591035	0.0295517	−	0.999563i	$-0.490592\pi$
$797$	1.66856	0.0591035	0.0295517	+	0.999563i	$0.490592\pi$
$798$	0	0
$799$	93.4994	3.30777
$800$	0	0
$801$	10.8367	0.382896
$802$	0	0
$803$	−15.2086	−0.536699
$804$	0	0
$805$	−4.66596	−0.164453
$806$	0	0
$807$	−28.1232	−0.989982
$808$	0	0
$809$	−9.91277	−0.348515	−0.174257	−	0.984700i	$-0.555752\pi$
$809$	−9.91277	−0.348515	−0.174257	+	0.984700i	$0.555752\pi$
$810$	0	0
$811$	41.7942	1.46759	0.733797	−	0.679369i	$-0.237747\pi$
$811$	41.7942	1.46759	0.733797	+	0.679369i	$0.237747\pi$
$812$	0	0
$813$	27.3764	0.960132
$814$	0	0
$815$	−4.47673	−0.156813
$816$	0	0
$817$	33.0917	1.15773
$818$	0	0
$819$	8.10114	0.283077
$820$	0	0
$821$	46.0792	1.60818	0.804088	−	0.594511i	$-0.202654\pi$
$821$	46.0792	1.60818	0.804088	+	0.594511i	$0.202654\pi$
$822$	0	0
$823$	28.1828	0.982392	0.491196	−	0.871049i	$-0.336560\pi$
$823$	28.1828	0.982392	0.491196	+	0.871049i	$0.336560\pi$
$824$	0	0
$825$	−7.39478	−0.257453
$826$	0	0
$827$	0.268104	0.00932290	0.00466145	−	0.999989i	$-0.498516\pi$
$827$	0.268104	0.00932290	0.00466145	+	0.999989i	$0.498516\pi$
$828$	0	0
$829$	−18.1618	−0.630787	−0.315393	−	0.948961i	$-0.602136\pi$
$829$	−18.1618	−0.630787	−0.315393	+	0.948961i	$0.602136\pi$
$830$	0	0
$831$	10.8716	0.377133
$832$	0	0
$833$	−10.2617	−0.355548
$834$	0	0
$835$	0.0176075	0.000609333 0
$836$	0	0
$837$	−32.1549	−1.11144
$838$	0	0
$839$	9.35795	0.323072	0.161536	−	0.986867i	$-0.448355\pi$
$839$	9.35795	0.323072	0.161536	+	0.986867i	$0.448355\pi$
$840$	0	0
$841$	−21.9222	−0.755938
$842$	0	0
$843$	24.3344	0.838121
$844$	0	0
$845$	3.67778	0.126520
$846$	0	0
$847$	−23.8753	−0.820364
$848$	0	0
$849$	24.4120	0.837819
$850$	0	0
$851$	1.94069	0.0665261
$852$	0	0
$853$	−6.02773	−0.206386	−0.103193	−	0.994661i	$-0.532906\pi$
$853$	−6.02773	−0.206386	−0.103193	+	0.994661i	$0.532906\pi$
$854$	0	0
$855$	−0.858302	−0.0293533
$856$	0	0
$857$	10.0052	0.341770	0.170885	−	0.985291i	$-0.445337\pi$
$857$	10.0052	0.341770	0.170885	+	0.985291i	$0.445337\pi$
$858$	0	0
$859$	−51.9917	−1.77394	−0.886968	−	0.461832i	$-0.847192\pi$
$859$	−51.9917	−1.77394	−0.886968	+	0.461832i	$0.847192\pi$
$860$	0	0
$861$	15.0184	0.511824
$862$	0	0
$863$	19.4576	0.662344	0.331172	−	0.943570i	$-0.392556\pi$
$863$	19.4576	0.662344	0.331172	+	0.943570i	$0.392556\pi$
$864$	0	0
$865$	6.59335	0.224181
$866$	0	0
$867$	−63.8143	−2.16725
$868$	0	0
$869$	11.9510	0.405409
$870$	0	0
$871$	54.7151	1.85395
$872$	0	0
$873$	−4.40539	−0.149100
$874$	0	0
$875$	−6.52483	−0.220580
$876$	0	0
$877$	38.0330	1.28428	0.642142	−	0.766586i	$-0.278046\pi$
$877$	38.0330	1.28428	0.642142	+	0.766586i	$0.278046\pi$
$878$	0	0
$879$	18.6385	0.628662
$880$	0	0
$881$	18.3218	0.617277	0.308638	−	0.951179i	$-0.400127\pi$
$881$	18.3218	0.617277	0.308638	+	0.951179i	$0.400127\pi$
$882$	0	0
$883$	−44.2160	−1.48799	−0.743993	−	0.668187i	$-0.767071\pi$
$883$	−44.2160	−1.48799	−0.743993	+	0.668187i	$0.767071\pi$
$884$	0	0
$885$	2.87298	0.0965740
$886$	0	0
$887$	35.6691	1.19765	0.598825	−	0.800880i	$-0.295634\pi$
$887$	35.6691	1.19765	0.598825	+	0.800880i	$0.295634\pi$
$888$	0	0
$889$	4.02176	0.134885
$890$	0	0
$891$	6.45379	0.216210
$892$	0	0
$893$	57.0765	1.90999
$894$	0	0
$895$	−3.95056	−0.132053
$896$	0	0
$897$	−55.6350	−1.85760
$898$	0	0
$899$	−15.2760	−0.509483
$900$	0	0
$901$	−5.57185	−0.185625
$902$	0	0
$903$	−25.7201	−0.855912
$904$	0	0
$905$	4.63205	0.153975
$906$	0	0
$907$	−26.6991	−0.886530	−0.443265	−	0.896391i	$-0.646180\pi$
$907$	−26.6991	−0.886530	−0.443265	+	0.896391i	$0.646180\pi$
$908$	0	0
$909$	−6.86417	−0.227670
$910$	0	0
$911$	52.9433	1.75409	0.877045	−	0.480408i	$-0.159511\pi$
$911$	52.9433	1.75409	0.877045	+	0.480408i	$0.159511\pi$
$912$	0	0
$913$	8.41542	0.278510
$914$	0	0
$915$	−2.52768	−0.0835625
$916$	0	0
$917$	26.9382	0.889577
$918$	0	0
$919$	6.37609	0.210328	0.105164	−	0.994455i	$-0.466463\pi$
$919$	6.37609	0.210328	0.105164	+	0.994455i	$0.466463\pi$
$920$	0	0
$921$	39.6832	1.30761
$922$	0	0
$923$	42.2356	1.39020
$924$	0	0
$925$	1.34648	0.0442720
$926$	0	0
$927$	−1.76699	−0.0580356
$928$	0	0
$929$	15.7445	0.516559	0.258279	−	0.966070i	$-0.416845\pi$
$929$	15.7445	0.516559	0.258279	+	0.966070i	$0.416845\pi$
$930$	0	0
$931$	−6.26426	−0.205303
$932$	0	0
$933$	15.1754	0.496822
$934$	0	0
$935$	−2.08130	−0.0680659
$936$	0	0
$937$	−17.1060	−0.558827	−0.279414	−	0.960171i	$-0.590140\pi$
$937$	−17.1060	−0.558827	−0.279414	+	0.960171i	$0.590140\pi$
$938$	0	0
$939$	−34.3788	−1.12191
$940$	0	0
$941$	−2.71609	−0.0885421	−0.0442710	−	0.999020i	$-0.514097\pi$
$941$	−2.71609	−0.0885421	−0.0442710	+	0.999020i	$0.514097\pi$
$942$	0	0
$943$	29.3060	0.954334
$944$	0	0
$945$	3.68201	0.119776
$946$	0	0
$947$	−44.7796	−1.45514	−0.727571	−	0.686032i	$-0.759351\pi$
$947$	−44.7796	−1.45514	−0.727571	+	0.686032i	$0.759351\pi$
$948$	0	0
$949$	−79.3880	−2.57704
$950$	0	0
$951$	9.42892	0.305754
$952$	0	0
$953$	−10.3455	−0.335123	−0.167562	−	0.985862i	$-0.553589\pi$
$953$	−10.3455	−0.335123	−0.167562	+	0.985862i	$0.553589\pi$
$954$	0	0
$955$	−4.11912	−0.133292
$956$	0	0
$957$	3.99566	0.129161
$958$	0	0
$959$	−20.6336	−0.666295
$960$	0	0
$961$	1.97015	0.0635533
$962$	0	0
$963$	6.17085	0.198853
$964$	0	0
$965$	−1.24793	−0.0401721
$966$	0	0
$967$	21.7745	0.700222	0.350111	−	0.936708i	$-0.386144\pi$
$967$	21.7745	0.700222	0.350111	+	0.936708i	$0.386144\pi$
$968$	0	0
$969$	−54.8171	−1.76098
$970$	0	0
$971$	−2.57418	−0.0826093	−0.0413046	−	0.999147i	$-0.513151\pi$
$971$	−2.57418	−0.0826093	−0.0413046	+	0.999147i	$0.513151\pi$
$972$	0	0
$973$	34.8958	1.11871
$974$	0	0
$975$	−38.6004	−1.23620
$976$	0	0
$977$	−1.36961	−0.0438178	−0.0219089	−	0.999760i	$-0.506974\pi$
$977$	−1.36961	−0.0438178	−0.0219089	+	0.999760i	$0.506974\pi$
$978$	0	0
$979$	16.0414	0.512685
$980$	0	0
$981$	−7.31425	−0.233526
$982$	0	0
$983$	10.2147	0.325799	0.162899	−	0.986643i	$-0.447915\pi$
$983$	10.2147	0.325799	0.162899	+	0.986643i	$0.447915\pi$
$984$	0	0
$985$	4.96148	0.158086
$986$	0	0
$987$	−44.3620	−1.41206
$988$	0	0
$989$	−50.1888	−1.59591
$990$	0	0
$991$	−18.2800	−0.580682	−0.290341	−	0.956923i	$-0.593769\pi$
$991$	−18.2800	−0.580682	−0.290341	+	0.956923i	$0.593769\pi$
$992$	0	0
$993$	37.2397	1.18177
$994$	0	0
$995$	3.47872	0.110283
$996$	0	0
$997$	−13.3344	−0.422304	−0.211152	−	0.977453i	$-0.567721\pi$
$997$	−13.3344	−0.422304	−0.211152	+	0.977453i	$0.567721\pi$
$998$	0	0
$999$	−1.53144	−0.0484528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.16	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.16	✓	44	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)

\(f(q)\)	\(=\)	\(q-1.52846 q^{3} +0.276340 q^{5} +2.37933 q^{7} -0.663805 q^{9} +O(q^{10})\)	\(q-1.52846 q^{3} +0.276340 q^{5} +2.37933 q^{7} -0.663805 q^{9} -0.982619 q^{11} -5.12922 q^{13} -0.422376 q^{15} +7.66490 q^{17} +4.67902 q^{19} -3.63671 q^{21} -7.09647 q^{23} -4.92364 q^{25} +5.59999 q^{27} +2.66042 q^{29} -5.74196 q^{31} +1.50189 q^{33} +0.657504 q^{35} -0.273473 q^{37} +7.83981 q^{39} -4.12965 q^{41} +7.07236 q^{43} -0.183436 q^{45} +12.1984 q^{47} -1.33880 q^{49} -11.7155 q^{51} -0.726931 q^{53} -0.271537 q^{55} -7.15171 q^{57} -6.80195 q^{59} +5.98444 q^{61} -1.57941 q^{63} -1.41741 q^{65} -10.6673 q^{67} +10.8467 q^{69} -8.23431 q^{71} +15.4776 q^{73} +7.52559 q^{75} -2.33797 q^{77} -12.1624 q^{79} -6.56795 q^{81} -8.56428 q^{83} +2.11812 q^{85} -4.06634 q^{87} -16.3251 q^{89} -12.2041 q^{91} +8.77637 q^{93} +1.29300 q^{95} +6.63657 q^{97} +0.652267 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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