LMFDB - Embedded newform 6008.2.a.d.1.17

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:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q-1.04405 q^{3} -3.83943 q^{5} +4.18991 q^{7} -1.90997 q^{9} +O(q^{10})$	$q-1.04405 q^{3} -3.83943 q^{5} +4.18991 q^{7} -1.90997 q^{9} +2.25887 q^{11} +6.21540 q^{13} +4.00855 q^{15} +3.58298 q^{17} -2.36756 q^{19} -4.37446 q^{21} +8.02816 q^{23} +9.74125 q^{25} +5.12623 q^{27} +1.06363 q^{29} +0.0251727 q^{31} -2.35837 q^{33} -16.0869 q^{35} +0.0470689 q^{37} -6.48916 q^{39} -2.00782 q^{41} -6.19535 q^{43} +7.33319 q^{45} +3.62361 q^{47} +10.5553 q^{49} -3.74080 q^{51} +1.90179 q^{53} -8.67278 q^{55} +2.47184 q^{57} -10.5245 q^{59} +2.29071 q^{61} -8.00258 q^{63} -23.8636 q^{65} -7.85627 q^{67} -8.38178 q^{69} -10.6765 q^{71} +5.05756 q^{73} -10.1703 q^{75} +9.46446 q^{77} +14.2043 q^{79} +0.377873 q^{81} +0.601026 q^{83} -13.7566 q^{85} -1.11048 q^{87} +0.230410 q^{89} +26.0419 q^{91} -0.0262815 q^{93} +9.09007 q^{95} -1.85008 q^{97} -4.31437 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * 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
$3$	−1.04405	−0.602781	−0.301390	−	0.953501i	$-0.597451\pi$
$3$	−1.04405	−0.602781	−0.301390	+	0.953501i	$0.597451\pi$
$4$	0	0
$5$	−3.83943	−1.71705	−0.858523	−	0.512775i	$-0.828618\pi$
$5$	−3.83943	−1.71705	−0.858523	+	0.512775i	$0.828618\pi$
$6$	0	0
$7$	4.18991	1.58364	0.791818	−	0.610757i	$-0.209135\pi$
$7$	4.18991	1.58364	0.791818	+	0.610757i	$0.209135\pi$
$8$	0	0
$9$	−1.90997	−0.636656
$10$	0	0
$11$	2.25887	0.681075	0.340538	−	0.940231i	$-0.389391\pi$
$11$	2.25887	0.681075	0.340538	+	0.940231i	$0.389391\pi$
$12$	0	0
$13$	6.21540	1.72384	0.861921	−	0.507043i	$-0.169262\pi$
$13$	6.21540	1.72384	0.861921	+	0.507043i	$0.169262\pi$
$14$	0	0
$15$	4.00855	1.03500
$16$	0	0
$17$	3.58298	0.869000	0.434500	−	0.900672i	$-0.356925\pi$
$17$	3.58298	0.869000	0.434500	+	0.900672i	$0.356925\pi$
$18$	0	0
$19$	−2.36756	−0.543155	−0.271577	−	0.962417i	$-0.587545\pi$
$19$	−2.36756	−0.543155	−0.271577	+	0.962417i	$0.587545\pi$
$20$	0	0
$21$	−4.37446	−0.954585
$22$	0	0
$23$	8.02816	1.67399	0.836994	−	0.547212i	$-0.184311\pi$
$23$	8.02816	1.67399	0.836994	+	0.547212i	$0.184311\pi$
$24$	0	0
$25$	9.74125	1.94825
$26$	0	0
$27$	5.12623	0.986544
$28$	0	0
$29$	1.06363	0.197511	0.0987554	−	0.995112i	$-0.468514\pi$
$29$	1.06363	0.197511	0.0987554	+	0.995112i	$0.468514\pi$
$30$	0	0
$31$	0.0251727	0.00452115	0.00226057	−	0.999997i	$-0.499280\pi$
$31$	0.0251727	0.00452115	0.00226057	+	0.999997i	$0.499280\pi$
$32$	0	0
$33$	−2.35837	−0.410539
$34$	0	0
$35$	−16.0869	−2.71918
$36$	0	0
$37$	0.0470689	0.00773808	0.00386904	−	0.999993i	$-0.498768\pi$
$37$	0.0470689	0.00773808	0.00386904	+	0.999993i	$0.498768\pi$
$38$	0	0
$39$	−6.48916	−1.03910
$40$	0	0
$41$	−2.00782	−0.313569	−0.156785	−	0.987633i	$-0.550113\pi$
$41$	−2.00782	−0.313569	−0.156785	+	0.987633i	$0.550113\pi$
$42$	0	0
$43$	−6.19535	−0.944782	−0.472391	−	0.881389i	$-0.656609\pi$
$43$	−6.19535	−0.944782	−0.472391	+	0.881389i	$0.656609\pi$
$44$	0	0
$45$	7.33319	1.09317
$46$	0	0
$47$	3.62361	0.528558	0.264279	−	0.964446i	$-0.414866\pi$
$47$	3.62361	0.528558	0.264279	+	0.964446i	$0.414866\pi$
$48$	0	0
$49$	10.5553	1.50790
$50$	0	0
$51$	−3.74080	−0.523816
$52$	0	0
$53$	1.90179	0.261230	0.130615	−	0.991433i	$-0.458305\pi$
$53$	1.90179	0.261230	0.130615	+	0.991433i	$0.458305\pi$
$54$	0	0
$55$	−8.67278	−1.16944
$56$	0	0
$57$	2.47184	0.327403
$58$	0	0
$59$	−10.5245	−1.37017	−0.685084	−	0.728464i	$-0.740234\pi$
$59$	−10.5245	−1.37017	−0.685084	+	0.728464i	$0.740234\pi$
$60$	0	0
$61$	2.29071	0.293296	0.146648	−	0.989189i	$-0.453152\pi$
$61$	2.29071	0.293296	0.146648	+	0.989189i	$0.453152\pi$
$62$	0	0
$63$	−8.00258	−1.00823
$64$	0	0
$65$	−23.8636	−2.95992
$66$	0	0
$67$	−7.85627	−0.959796	−0.479898	−	0.877324i	$-0.659326\pi$
$67$	−7.85627	−0.959796	−0.479898	+	0.877324i	$0.659326\pi$
$68$	0	0
$69$	−8.38178	−1.00905
$70$	0	0
$71$	−10.6765	−1.26707	−0.633534	−	0.773714i	$-0.718397\pi$
$71$	−10.6765	−1.26707	−0.633534	+	0.773714i	$0.718397\pi$
$72$	0	0
$73$	5.05756	0.591943	0.295971	−	0.955197i	$-0.404357\pi$
$73$	5.05756	0.591943	0.295971	+	0.955197i	$0.404357\pi$
$74$	0	0
$75$	−10.1703	−1.17437
$76$	0	0
$77$	9.46446	1.07858
$78$	0	0
$79$	14.2043	1.59811	0.799055	−	0.601258i	$-0.205334\pi$
$79$	14.2043	1.59811	0.799055	+	0.601258i	$0.205334\pi$
$80$	0	0
$81$	0.377873	0.0419859
$82$	0	0
$83$	0.601026	0.0659712	0.0329856	−	0.999456i	$-0.489498\pi$
$83$	0.601026	0.0659712	0.0329856	+	0.999456i	$0.489498\pi$
$84$	0	0
$85$	−13.7566	−1.49211
$86$	0	0
$87$	−1.11048	−0.119056
$88$	0	0
$89$	0.230410	0.0244234	0.0122117	−	0.999925i	$-0.496113\pi$
$89$	0.230410	0.0244234	0.0122117	+	0.999925i	$0.496113\pi$
$90$	0	0
$91$	26.0419	2.72994
$92$	0	0
$93$	−0.0262815	−0.00272526
$94$	0	0
$95$	9.09007	0.932622
$96$	0	0
$97$	−1.85008	−0.187848	−0.0939238	−	0.995579i	$-0.529941\pi$
$97$	−1.85008	−0.187848	−0.0939238	+	0.995579i	$0.529941\pi$
$98$	0	0
$99$	−4.31437	−0.433610
$100$	0	0
$101$	8.32344	0.828213	0.414107	−	0.910228i	$-0.364094\pi$
$101$	8.32344	0.828213	0.414107	+	0.910228i	$0.364094\pi$
$102$	0	0
$103$	−16.1655	−1.59283	−0.796415	−	0.604751i	$-0.793273\pi$
$103$	−16.1655	−1.59283	−0.796415	+	0.604751i	$0.793273\pi$
$104$	0	0
$105$	16.7954	1.63907
$106$	0	0
$107$	−5.65380	−0.546573	−0.273287	−	0.961933i	$-0.588111\pi$
$107$	−5.65380	−0.546573	−0.273287	+	0.961933i	$0.588111\pi$
$108$	0	0
$109$	16.8750	1.61633	0.808165	−	0.588956i	$-0.200461\pi$
$109$	16.8750	1.61633	0.808165	+	0.588956i	$0.200461\pi$
$110$	0	0
$111$	−0.0491421	−0.00466436
$112$	0	0
$113$	1.34748	0.126760	0.0633802	−	0.997989i	$-0.479812\pi$
$113$	1.34748	0.126760	0.0633802	+	0.997989i	$0.479812\pi$
$114$	0	0
$115$	−30.8236	−2.87431
$116$	0	0
$117$	−11.8712	−1.09749
$118$	0	0
$119$	15.0123	1.37618
$120$	0	0
$121$	−5.89750	−0.536137
$122$	0	0
$123$	2.09626	0.189013
$124$	0	0
$125$	−18.2037	−1.62819
$126$	0	0
$127$	15.5015	1.37554	0.687768	−	0.725931i	$-0.258591\pi$
$127$	15.5015	1.37554	0.687768	+	0.725931i	$0.258591\pi$
$128$	0	0
$129$	6.46823	0.569496
$130$	0	0
$131$	4.57980	0.400139	0.200069	−	0.979782i	$-0.435883\pi$
$131$	4.57980	0.400139	0.200069	+	0.979782i	$0.435883\pi$
$132$	0	0
$133$	−9.91984	−0.860159
$134$	0	0
$135$	−19.6818	−1.69394
$136$	0	0
$137$	−11.1046	−0.948726	−0.474363	−	0.880329i	$-0.657322\pi$
$137$	−11.1046	−0.948726	−0.474363	+	0.880329i	$0.657322\pi$
$138$	0	0
$139$	18.3832	1.55924	0.779620	−	0.626253i	$-0.215412\pi$
$139$	18.3832	1.55924	0.779620	+	0.626253i	$0.215412\pi$
$140$	0	0
$141$	−3.78322	−0.318605
$142$	0	0
$143$	14.0398	1.17407
$144$	0	0
$145$	−4.08373	−0.339135
$146$	0	0
$147$	−11.0202	−0.908935
$148$	0	0
$149$	−23.1586	−1.89722	−0.948612	−	0.316441i	$-0.897512\pi$
$149$	−23.1586	−1.89722	−0.948612	+	0.316441i	$0.897512\pi$
$150$	0	0
$151$	22.3235	1.81666	0.908330	−	0.418255i	$-0.137358\pi$
$151$	22.3235	1.81666	0.908330	+	0.418255i	$0.137358\pi$
$152$	0	0
$153$	−6.84337	−0.553253
$154$	0	0
$155$	−0.0966488	−0.00776302
$156$	0	0
$157$	12.9256	1.03157	0.515787	−	0.856717i	$-0.327500\pi$
$157$	12.9256	1.03157	0.515787	+	0.856717i	$0.327500\pi$
$158$	0	0
$159$	−1.98555	−0.157465
$160$	0	0
$161$	33.6373	2.65099
$162$	0	0
$163$	−7.49370	−0.586952	−0.293476	−	0.955966i	$-0.594812\pi$
$163$	−7.49370	−0.586952	−0.293476	+	0.955966i	$0.594812\pi$
$164$	0	0
$165$	9.05479	0.704914
$166$	0	0
$167$	8.93016	0.691036	0.345518	−	0.938412i	$-0.387703\pi$
$167$	8.93016	0.691036	0.345518	+	0.938412i	$0.387703\pi$
$168$	0	0
$169$	25.6312	1.97163
$170$	0	0
$171$	4.52195	0.345802
$172$	0	0
$173$	−24.1991	−1.83982	−0.919911	−	0.392128i	$-0.871739\pi$
$173$	−24.1991	−1.83982	−0.919911	+	0.392128i	$0.871739\pi$
$174$	0	0
$175$	40.8149	3.08532
$176$	0	0
$177$	10.9880	0.825910
$178$	0	0
$179$	0.802271	0.0599645	0.0299823	−	0.999550i	$-0.490455\pi$
$179$	0.802271	0.0599645	0.0299823	+	0.999550i	$0.490455\pi$
$180$	0	0
$181$	−16.8457	−1.25213	−0.626066	−	0.779770i	$-0.715336\pi$
$181$	−16.8457	−1.25213	−0.626066	+	0.779770i	$0.715336\pi$
$182$	0	0
$183$	−2.39161	−0.176793
$184$	0	0
$185$	−0.180718	−0.0132866
$186$	0	0
$187$	8.09348	0.591854
$188$	0	0
$189$	21.4784	1.56233
$190$	0	0
$191$	2.30869	0.167051	0.0835256	−	0.996506i	$-0.473382\pi$
$191$	2.30869	0.167051	0.0835256	+	0.996506i	$0.473382\pi$
$192$	0	0
$193$	6.63032	0.477261	0.238630	−	0.971110i	$-0.423302\pi$
$193$	6.63032	0.477261	0.238630	+	0.971110i	$0.423302\pi$
$194$	0	0
$195$	24.9147	1.78418
$196$	0	0
$197$	−6.54301	−0.466170	−0.233085	−	0.972456i	$-0.574882\pi$
$197$	−6.54301	−0.466170	−0.233085	+	0.972456i	$0.574882\pi$
$198$	0	0
$199$	7.82535	0.554724	0.277362	−	0.960765i	$-0.410540\pi$
$199$	7.82535	0.554724	0.277362	+	0.960765i	$0.410540\pi$
$200$	0	0
$201$	8.20231	0.578546
$202$	0	0
$203$	4.45650	0.312785
$204$	0	0
$205$	7.70890	0.538413
$206$	0	0
$207$	−15.3335	−1.06575
$208$	0	0
$209$	−5.34800	−0.369929
$210$	0	0
$211$	−18.8276	−1.29615	−0.648073	−	0.761578i	$-0.724425\pi$
$211$	−18.8276	−1.29615	−0.648073	+	0.761578i	$0.724425\pi$
$212$	0	0
$213$	11.1468	0.763765
$214$	0	0
$215$	23.7866	1.62223
$216$	0	0
$217$	0.105471	0.00715985
$218$	0	0
$219$	−5.28033	−0.356812
$220$	0	0
$221$	22.2696	1.49802
$222$	0	0
$223$	4.27081	0.285995	0.142997	−	0.989723i	$-0.454326\pi$
$223$	4.27081	0.285995	0.142997	+	0.989723i	$0.454326\pi$
$224$	0	0
$225$	−18.6055	−1.24036
$226$	0	0
$227$	7.01313	0.465478	0.232739	−	0.972539i	$-0.425231\pi$
$227$	7.01313	0.465478	0.232739	+	0.972539i	$0.425231\pi$
$228$	0	0
$229$	25.8559	1.70861	0.854303	−	0.519775i	$-0.173984\pi$
$229$	25.8559	1.70861	0.854303	+	0.519775i	$0.173984\pi$
$230$	0	0
$231$	−9.88134	−0.650144
$232$	0	0
$233$	−2.94855	−0.193166	−0.0965829	−	0.995325i	$-0.530791\pi$
$233$	−2.94855	−0.193166	−0.0965829	+	0.995325i	$0.530791\pi$
$234$	0	0
$235$	−13.9126	−0.907559
$236$	0	0
$237$	−14.8300	−0.963310
$238$	0	0
$239$	−26.4256	−1.70933	−0.854667	−	0.519177i	$-0.826238\pi$
$239$	−26.4256	−1.70933	−0.854667	+	0.519177i	$0.826238\pi$
$240$	0	0
$241$	17.8731	1.15131	0.575654	−	0.817693i	$-0.304747\pi$
$241$	17.8731	1.15131	0.575654	+	0.817693i	$0.304747\pi$
$242$	0	0
$243$	−15.7732	−1.01185
$244$	0	0
$245$	−40.5265	−2.58914
$246$	0	0
$247$	−14.7153	−0.936312
$248$	0	0
$249$	−0.627499	−0.0397661
$250$	0	0
$251$	27.5799	1.74083	0.870413	−	0.492322i	$-0.163852\pi$
$251$	27.5799	1.74083	0.870413	+	0.492322i	$0.163852\pi$
$252$	0	0
$253$	18.1346	1.14011
$254$	0	0
$255$	14.3625	0.899417
$256$	0	0
$257$	9.14892	0.570694	0.285347	−	0.958424i	$-0.407891\pi$
$257$	9.14892	0.570694	0.285347	+	0.958424i	$0.407891\pi$
$258$	0	0
$259$	0.197214	0.0122543
$260$	0	0
$261$	−2.03149	−0.125746
$262$	0	0
$263$	11.8899	0.733161	0.366580	−	0.930386i	$-0.380528\pi$
$263$	11.8899	0.733161	0.366580	+	0.930386i	$0.380528\pi$
$264$	0	0
$265$	−7.30178	−0.448545
$266$	0	0
$267$	−0.240559	−0.0147219
$268$	0	0
$269$	2.93309	0.178834	0.0894169	−	0.995994i	$-0.471500\pi$
$269$	2.93309	0.178834	0.0894169	+	0.995994i	$0.471500\pi$
$270$	0	0
$271$	−12.5203	−0.760552	−0.380276	−	0.924873i	$-0.624171\pi$
$271$	−12.5203	−0.760552	−0.380276	+	0.924873i	$0.624171\pi$
$272$	0	0
$273$	−27.1890	−1.64555
$274$	0	0
$275$	22.0042	1.32690
$276$	0	0
$277$	−1.33972	−0.0804958	−0.0402479	−	0.999190i	$-0.512815\pi$
$277$	−1.33972	−0.0804958	−0.0402479	+	0.999190i	$0.512815\pi$
$278$	0	0
$279$	−0.0480790	−0.00287841
$280$	0	0
$281$	6.40424	0.382045	0.191022	−	0.981586i	$-0.438820\pi$
$281$	6.40424	0.382045	0.191022	+	0.981586i	$0.438820\pi$
$282$	0	0
$283$	−25.5168	−1.51682	−0.758409	−	0.651779i	$-0.774023\pi$
$283$	−25.5168	−1.51682	−0.758409	+	0.651779i	$0.774023\pi$
$284$	0	0
$285$	−9.49046	−0.562166
$286$	0	0
$287$	−8.41259	−0.496579
$288$	0	0
$289$	−4.16227	−0.244840
$290$	0	0
$291$	1.93157	0.113231
$292$	0	0
$293$	6.71815	0.392478	0.196239	−	0.980556i	$-0.437127\pi$
$293$	6.71815	0.392478	0.196239	+	0.980556i	$0.437127\pi$
$294$	0	0
$295$	40.4079	2.35264
$296$	0	0
$297$	11.5795	0.671911
$298$	0	0
$299$	49.8982	2.88569
$300$	0	0
$301$	−25.9579	−1.49619
$302$	0	0
$303$	−8.69006	−0.499231
$304$	0	0
$305$	−8.79504	−0.503603
$306$	0	0
$307$	19.7688	1.12826	0.564132	−	0.825685i	$-0.309211\pi$
$307$	19.7688	1.12826	0.564132	+	0.825685i	$0.309211\pi$
$308$	0	0
$309$	16.8775	0.960127
$310$	0	0
$311$	7.84751	0.444992	0.222496	−	0.974934i	$-0.428580\pi$
$311$	7.84751	0.444992	0.222496	+	0.974934i	$0.428580\pi$
$312$	0	0
$313$	5.61187	0.317202	0.158601	−	0.987343i	$-0.449302\pi$
$313$	5.61187	0.317202	0.158601	+	0.987343i	$0.449302\pi$
$314$	0	0
$315$	30.7254	1.73118
$316$	0	0
$317$	−32.2715	−1.81255	−0.906273	−	0.422692i	$-0.861085\pi$
$317$	−32.2715	−1.81255	−0.906273	+	0.422692i	$0.861085\pi$
$318$	0	0
$319$	2.40260	0.134520
$320$	0	0
$321$	5.90283	0.329464
$322$	0	0
$323$	−8.48290	−0.472001
$324$	0	0
$325$	60.5457	3.35847
$326$	0	0
$327$	−17.6183	−0.974293
$328$	0	0
$329$	15.1826	0.837044
$330$	0	0
$331$	22.1847	1.21938	0.609692	−	0.792638i	$-0.291293\pi$
$331$	22.1847	1.21938	0.609692	+	0.792638i	$0.291293\pi$
$332$	0	0
$333$	−0.0899000	−0.00492649
$334$	0	0
$335$	30.1636	1.64801
$336$	0	0
$337$	−33.7030	−1.83592	−0.917959	−	0.396674i	$-0.870164\pi$
$337$	−33.7030	−1.83592	−0.917959	+	0.396674i	$0.870164\pi$
$338$	0	0
$339$	−1.40683	−0.0764087
$340$	0	0
$341$	0.0568618	0.00307924
$342$	0	0
$343$	14.8965	0.804334
$344$	0	0
$345$	32.1813	1.73258
$346$	0	0
$347$	11.8042	0.633683	0.316841	−	0.948479i	$-0.397378\pi$
$347$	11.8042	0.633683	0.316841	+	0.948479i	$0.397378\pi$
$348$	0	0
$349$	19.8168	1.06077	0.530385	−	0.847757i	$-0.322047\pi$
$349$	19.8168	1.06077	0.530385	+	0.847757i	$0.322047\pi$
$350$	0	0
$351$	31.8616	1.70065
$352$	0	0
$353$	−23.8357	−1.26865	−0.634323	−	0.773068i	$-0.718721\pi$
$353$	−23.8357	−1.26865	−0.634323	+	0.773068i	$0.718721\pi$
$354$	0	0
$355$	40.9918	2.17562
$356$	0	0
$357$	−15.6736	−0.829534
$358$	0	0
$359$	18.1196	0.956316	0.478158	−	0.878274i	$-0.341305\pi$
$359$	18.1196	0.956316	0.478158	+	0.878274i	$0.341305\pi$
$360$	0	0
$361$	−13.3947	−0.704983
$362$	0	0
$363$	6.15727	0.323173
$364$	0	0
$365$	−19.4182	−1.01639
$366$	0	0
$367$	27.3661	1.42850	0.714249	−	0.699892i	$-0.246769\pi$
$367$	27.3661	1.42850	0.714249	+	0.699892i	$0.246769\pi$
$368$	0	0
$369$	3.83487	0.199636
$370$	0	0
$371$	7.96830	0.413694
$372$	0	0
$373$	20.0859	1.04001	0.520005	−	0.854163i	$-0.325930\pi$
$373$	20.0859	1.04001	0.520005	+	0.854163i	$0.325930\pi$
$374$	0	0
$375$	19.0055	0.981440
$376$	0	0
$377$	6.61087	0.340477
$378$	0	0
$379$	9.32944	0.479221	0.239610	−	0.970869i	$-0.422980\pi$
$379$	9.32944	0.479221	0.239610	+	0.970869i	$0.422980\pi$
$380$	0	0
$381$	−16.1843	−0.829146
$382$	0	0
$383$	18.7243	0.956769	0.478385	−	0.878150i	$-0.341222\pi$
$383$	18.7243	0.956769	0.478385	+	0.878150i	$0.341222\pi$
$384$	0	0
$385$	−36.3382	−1.85196
$386$	0	0
$387$	11.8329	0.601501
$388$	0	0
$389$	14.2239	0.721180	0.360590	−	0.932724i	$-0.382575\pi$
$389$	14.2239	0.721180	0.360590	+	0.932724i	$0.382575\pi$
$390$	0	0
$391$	28.7647	1.45469
$392$	0	0
$393$	−4.78152	−0.241196
$394$	0	0
$395$	−54.5365	−2.74403
$396$	0	0
$397$	33.5861	1.68564	0.842820	−	0.538196i	$-0.180894\pi$
$397$	33.5861	1.68564	0.842820	+	0.538196i	$0.180894\pi$
$398$	0	0
$399$	10.3568	0.518487
$400$	0	0
$401$	26.6840	1.33253	0.666267	−	0.745714i	$-0.267891\pi$
$401$	26.6840	1.33253	0.666267	+	0.745714i	$0.267891\pi$
$402$	0	0
$403$	0.156458	0.00779374
$404$	0	0
$405$	−1.45082	−0.0720917
$406$	0	0
$407$	0.106322	0.00527021
$408$	0	0
$409$	3.68152	0.182040	0.0910198	−	0.995849i	$-0.470987\pi$
$409$	3.68152	0.182040	0.0910198	+	0.995849i	$0.470987\pi$
$410$	0	0
$411$	11.5937	0.571873
$412$	0	0
$413$	−44.0965	−2.16985
$414$	0	0
$415$	−2.30760	−0.113276
$416$	0	0
$417$	−19.1929	−0.939879
$418$	0	0
$419$	−39.9143	−1.94994	−0.974970	−	0.222338i	$-0.928631\pi$
$419$	−39.9143	−1.94994	−0.974970	+	0.222338i	$0.928631\pi$
$420$	0	0
$421$	−8.35874	−0.407380	−0.203690	−	0.979035i	$-0.565293\pi$
$421$	−8.35874	−0.407380	−0.203690	+	0.979035i	$0.565293\pi$
$422$	0	0
$423$	−6.92098	−0.336510
$424$	0	0
$425$	34.9027	1.69303
$426$	0	0
$427$	9.59788	0.464474
$428$	0	0
$429$	−14.6582	−0.707704
$430$	0	0
$431$	−8.26785	−0.398248	−0.199124	−	0.979974i	$-0.563810\pi$
$431$	−8.26785	−0.398248	−0.199124	+	0.979974i	$0.563810\pi$
$432$	0	0
$433$	1.26347	0.0607186	0.0303593	−	0.999539i	$-0.490335\pi$
$433$	1.26347	0.0607186	0.0303593	+	0.999539i	$0.490335\pi$
$434$	0	0
$435$	4.26360	0.204424
$436$	0	0
$437$	−19.0071	−0.909234
$438$	0	0
$439$	−9.28699	−0.443244	−0.221622	−	0.975133i	$-0.571135\pi$
$439$	−9.28699	−0.443244	−0.221622	+	0.975133i	$0.571135\pi$
$440$	0	0
$441$	−20.1603	−0.960015
$442$	0	0
$443$	−4.72362	−0.224426	−0.112213	−	0.993684i	$-0.535794\pi$
$443$	−4.72362	−0.224426	−0.112213	+	0.993684i	$0.535794\pi$
$444$	0	0
$445$	−0.884643	−0.0419361
$446$	0	0
$447$	24.1786	1.14361
$448$	0	0
$449$	−13.2153	−0.623668	−0.311834	−	0.950137i	$-0.600943\pi$
$449$	−13.2153	−0.623668	−0.311834	+	0.950137i	$0.600943\pi$
$450$	0	0
$451$	−4.53541	−0.213564
$452$	0	0
$453$	−23.3068	−1.09505
$454$	0	0
$455$	−99.9863	−4.68743
$456$	0	0
$457$	30.1327	1.40955	0.704775	−	0.709431i	$-0.251048\pi$
$457$	30.1327	1.40955	0.704775	+	0.709431i	$0.251048\pi$
$458$	0	0
$459$	18.3672	0.857307
$460$	0	0
$461$	−21.6939	−1.01038	−0.505192	−	0.863007i	$-0.668578\pi$
$461$	−21.6939	−1.01038	−0.505192	+	0.863007i	$0.668578\pi$
$462$	0	0
$463$	32.3025	1.50123	0.750613	−	0.660743i	$-0.229759\pi$
$463$	32.3025	1.50123	0.750613	+	0.660743i	$0.229759\pi$
$464$	0	0
$465$	0.100906	0.00467940
$466$	0	0
$467$	1.25917	0.0582673	0.0291336	−	0.999576i	$-0.490725\pi$
$467$	1.25917	0.0582673	0.0291336	+	0.999576i	$0.490725\pi$
$468$	0	0
$469$	−32.9170	−1.51997
$470$	0	0
$471$	−13.4949	−0.621813
$472$	0	0
$473$	−13.9945	−0.643467
$474$	0	0
$475$	−23.0629	−1.05820
$476$	0	0
$477$	−3.63235	−0.166314
$478$	0	0
$479$	4.40472	0.201257	0.100628	−	0.994924i	$-0.467915\pi$
$479$	4.40472	0.201257	0.100628	+	0.994924i	$0.467915\pi$
$480$	0	0
$481$	0.292552	0.0133392
$482$	0	0
$483$	−35.1189	−1.59796
$484$	0	0
$485$	7.10328	0.322543
$486$	0	0
$487$	−9.66635	−0.438024	−0.219012	−	0.975722i	$-0.570283\pi$
$487$	−9.66635	−0.438024	−0.219012	+	0.975722i	$0.570283\pi$
$488$	0	0
$489$	7.82378	0.353803
$490$	0	0
$491$	24.6433	1.11214	0.556068	−	0.831137i	$-0.312309\pi$
$491$	24.6433	1.11214	0.556068	+	0.831137i	$0.312309\pi$
$492$	0	0
$493$	3.81096	0.171637
$494$	0	0
$495$	16.5647	0.744529
$496$	0	0
$497$	−44.7336	−2.00658
$498$	0	0
$499$	9.40940	0.421223	0.210611	−	0.977570i	$-0.432455\pi$
$499$	9.40940	0.421223	0.210611	+	0.977570i	$0.432455\pi$
$500$	0	0
$501$	−9.32350	−0.416543
$502$	0	0
$503$	−36.8068	−1.64113	−0.820567	−	0.571551i	$-0.806342\pi$
$503$	−36.8068	−1.64113	−0.820567	+	0.571551i	$0.806342\pi$
$504$	0	0
$505$	−31.9573	−1.42208
$506$	0	0
$507$	−26.7601	−1.18846
$508$	0	0
$509$	12.3071	0.545504	0.272752	−	0.962084i	$-0.412066\pi$
$509$	12.3071	0.545504	0.272752	+	0.962084i	$0.412066\pi$
$510$	0	0
$511$	21.1907	0.937422
$512$	0	0
$513$	−12.1366	−0.535846
$514$	0	0
$515$	62.0662	2.73496
$516$	0	0
$517$	8.18527	0.359988
$518$	0	0
$519$	25.2650	1.10901
$520$	0	0
$521$	−32.7764	−1.43596	−0.717980	−	0.696064i	$-0.754933\pi$
$521$	−32.7764	−1.43596	−0.717980	+	0.696064i	$0.754933\pi$
$522$	0	0
$523$	20.3036	0.887816	0.443908	−	0.896072i	$-0.353592\pi$
$523$	20.3036	0.887816	0.443908	+	0.896072i	$0.353592\pi$
$524$	0	0
$525$	−42.6127	−1.85977
$526$	0	0
$527$	0.0901932	0.00392888
$528$	0	0
$529$	41.4514	1.80223
$530$	0	0
$531$	20.1014	0.872325
$532$	0	0
$533$	−12.4794	−0.540543
$534$	0	0
$535$	21.7074	0.938492
$536$	0	0
$537$	−0.837608	−0.0361455
$538$	0	0
$539$	23.8431	1.02700
$540$	0	0
$541$	−39.1705	−1.68407	−0.842036	−	0.539422i	$-0.818643\pi$
$541$	−39.1705	−1.68407	−0.842036	+	0.539422i	$0.818643\pi$
$542$	0	0
$543$	17.5877	0.754761
$544$	0	0
$545$	−64.7904	−2.77531
$546$	0	0
$547$	15.3606	0.656771	0.328385	−	0.944544i	$-0.393496\pi$
$547$	15.3606	0.656771	0.328385	+	0.944544i	$0.393496\pi$
$548$	0	0
$549$	−4.37519	−0.186728
$550$	0	0
$551$	−2.51820	−0.107279
$552$	0	0
$553$	59.5148	2.53082
$554$	0	0
$555$	0.188678	0.00800893
$556$	0	0
$557$	−32.9340	−1.39546	−0.697729	−	0.716362i	$-0.745806\pi$
$557$	−32.9340	−1.39546	−0.697729	+	0.716362i	$0.745806\pi$
$558$	0	0
$559$	−38.5066	−1.62865
$560$	0	0
$561$	−8.44997	−0.356758
$562$	0	0
$563$	−28.8421	−1.21555	−0.607775	−	0.794110i	$-0.707938\pi$
$563$	−28.8421	−1.21555	−0.607775	+	0.794110i	$0.707938\pi$
$564$	0	0
$565$	−5.17356	−0.217653
$566$	0	0
$567$	1.58325	0.0664903
$568$	0	0
$569$	−18.2439	−0.764826	−0.382413	−	0.923992i	$-0.624907\pi$
$569$	−18.2439	−0.764826	−0.382413	+	0.923992i	$0.624907\pi$
$570$	0	0
$571$	−2.11279	−0.0884174	−0.0442087	−	0.999022i	$-0.514077\pi$
$571$	−2.11279	−0.0884174	−0.0442087	+	0.999022i	$0.514077\pi$
$572$	0	0
$573$	−2.41038	−0.100695
$574$	0	0
$575$	78.2043	3.26134
$576$	0	0
$577$	30.5347	1.27118	0.635588	−	0.772028i	$-0.280758\pi$
$577$	30.5347	1.27118	0.635588	+	0.772028i	$0.280758\pi$
$578$	0	0
$579$	−6.92236	−0.287683
$580$	0	0
$581$	2.51824	0.104474
$582$	0	0
$583$	4.29589	0.177917
$584$	0	0
$585$	45.5787	1.88445
$586$	0	0
$587$	−20.2662	−0.836474	−0.418237	−	0.908338i	$-0.637352\pi$
$587$	−20.2662	−0.836474	−0.418237	+	0.908338i	$0.637352\pi$
$588$	0	0
$589$	−0.0595977	−0.00245568
$590$	0	0
$591$	6.83120	0.280998
$592$	0	0
$593$	1.20302	0.0494023	0.0247011	−	0.999695i	$-0.492137\pi$
$593$	1.20302	0.0494023	0.0247011	+	0.999695i	$0.492137\pi$
$594$	0	0
$595$	−57.6389	−2.36296
$596$	0	0
$597$	−8.17003	−0.334377
$598$	0	0
$599$	34.8594	1.42432	0.712158	−	0.702019i	$-0.247718\pi$
$599$	34.8594	1.42432	0.712158	+	0.702019i	$0.247718\pi$
$600$	0	0
$601$	−21.0171	−0.857304	−0.428652	−	0.903470i	$-0.641011\pi$
$601$	−21.0171	−0.857304	−0.428652	+	0.903470i	$0.641011\pi$
$602$	0	0
$603$	15.0052	0.611059
$604$	0	0
$605$	22.6431	0.920572
$606$	0	0
$607$	47.7233	1.93703	0.968515	−	0.248957i	$-0.0800876\pi$
$607$	47.7233	1.93703	0.968515	+	0.248957i	$0.0800876\pi$
$608$	0	0
$609$	−4.65280	−0.188541
$610$	0	0
$611$	22.5222	0.911151
$612$	0	0
$613$	11.1621	0.450833	0.225416	−	0.974263i	$-0.427626\pi$
$613$	11.1621	0.450833	0.225416	+	0.974263i	$0.427626\pi$
$614$	0	0
$615$	−8.04845	−0.324545
$616$	0	0
$617$	11.1158	0.447507	0.223753	−	0.974646i	$-0.428169\pi$
$617$	11.1158	0.447507	0.223753	+	0.974646i	$0.428169\pi$
$618$	0	0
$619$	−3.48423	−0.140043	−0.0700215	−	0.997545i	$-0.522307\pi$
$619$	−3.48423	−0.140043	−0.0700215	+	0.997545i	$0.522307\pi$
$620$	0	0
$621$	41.1542	1.65146
$622$	0	0
$623$	0.965396	0.0386778
$624$	0	0
$625$	21.1856	0.847425
$626$	0	0
$627$	5.58356	0.222986
$628$	0	0
$629$	0.168647	0.00672438
$630$	0	0
$631$	−0.876773	−0.0349038	−0.0174519	−	0.999848i	$-0.505555\pi$
$631$	−0.876773	−0.0349038	−0.0174519	+	0.999848i	$0.505555\pi$
$632$	0	0
$633$	19.6569	0.781291
$634$	0	0
$635$	−59.5170	−2.36186
$636$	0	0
$637$	65.6055	2.59939
$638$	0	0
$639$	20.3918	0.806686
$640$	0	0
$641$	43.0426	1.70008	0.850041	−	0.526717i	$-0.176577\pi$
$641$	43.0426	1.70008	0.850041	+	0.526717i	$0.176577\pi$
$642$	0	0
$643$	−0.299065	−0.0117940	−0.00589699	−	0.999983i	$-0.501877\pi$
$643$	−0.299065	−0.0117940	−0.00589699	+	0.999983i	$0.501877\pi$
$644$	0	0
$645$	−24.8343	−0.977852
$646$	0	0
$647$	−8.05727	−0.316764	−0.158382	−	0.987378i	$-0.550628\pi$
$647$	−8.05727	−0.316764	−0.158382	+	0.987378i	$0.550628\pi$
$648$	0	0
$649$	−23.7734	−0.933187
$650$	0	0
$651$	−0.110117	−0.00431582
$652$	0	0
$653$	−17.0801	−0.668397	−0.334199	−	0.942503i	$-0.608466\pi$
$653$	−17.0801	−0.668397	−0.334199	+	0.942503i	$0.608466\pi$
$654$	0	0
$655$	−17.5838	−0.687057
$656$	0	0
$657$	−9.65977	−0.376864
$658$	0	0
$659$	10.6373	0.414371	0.207186	−	0.978302i	$-0.433570\pi$
$659$	10.6373	0.414371	0.207186	+	0.978302i	$0.433570\pi$
$660$	0	0
$661$	4.31281	0.167749	0.0838744	−	0.996476i	$-0.473271\pi$
$661$	4.31281	0.167749	0.0838744	+	0.996476i	$0.473271\pi$
$662$	0	0
$663$	−23.2505	−0.902976
$664$	0	0
$665$	38.0866	1.47693
$666$	0	0
$667$	8.53898	0.330631
$668$	0	0
$669$	−4.45893	−0.172392
$670$	0	0
$671$	5.17443	0.199757
$672$	0	0
$673$	8.15228	0.314247	0.157124	−	0.987579i	$-0.449778\pi$
$673$	8.15228	0.314247	0.157124	+	0.987579i	$0.449778\pi$
$674$	0	0
$675$	49.9359	1.92203
$676$	0	0
$677$	45.5526	1.75073	0.875363	−	0.483466i	$-0.160622\pi$
$677$	45.5526	1.75073	0.875363	+	0.483466i	$0.160622\pi$
$678$	0	0
$679$	−7.75168	−0.297482
$680$	0	0
$681$	−7.32204	−0.280581
$682$	0	0
$683$	8.40356	0.321553	0.160777	−	0.986991i	$-0.448600\pi$
$683$	8.40356	0.321553	0.160777	+	0.986991i	$0.448600\pi$
$684$	0	0
$685$	42.6352	1.62901
$686$	0	0
$687$	−26.9948	−1.02992
$688$	0	0
$689$	11.8204	0.450320
$690$	0	0
$691$	−36.4217	−1.38555	−0.692773	−	0.721156i	$-0.743611\pi$
$691$	−36.4217	−1.38555	−0.692773	+	0.721156i	$0.743611\pi$
$692$	0	0
$693$	−18.0768	−0.686681
$694$	0	0
$695$	−70.5809	−2.67729
$696$	0	0
$697$	−7.19398	−0.272491
$698$	0	0
$699$	3.07842	0.116437
$700$	0	0
$701$	−19.0033	−0.717744	−0.358872	−	0.933387i	$-0.616838\pi$
$701$	−19.0033	−0.717744	−0.358872	+	0.933387i	$0.616838\pi$
$702$	0	0
$703$	−0.111438	−0.00420297
$704$	0	0
$705$	14.5254	0.547059
$706$	0	0
$707$	34.8744	1.31159
$708$	0	0
$709$	−27.4154	−1.02961	−0.514803	−	0.857309i	$-0.672135\pi$
$709$	−27.4154	−1.02961	−0.514803	+	0.857309i	$0.672135\pi$
$710$	0	0
$711$	−27.1298	−1.01745
$712$	0	0
$713$	0.202090	0.00756835
$714$	0	0
$715$	−53.9048	−2.01592
$716$	0	0
$717$	27.5896	1.03035
$718$	0	0
$719$	−20.3983	−0.760728	−0.380364	−	0.924837i	$-0.624201\pi$
$719$	−20.3983	−0.760728	−0.380364	+	0.924837i	$0.624201\pi$
$720$	0	0
$721$	−67.7318	−2.52246
$722$	0	0
$723$	−18.6604	−0.693986
$724$	0	0
$725$	10.3611	0.384800
$726$	0	0
$727$	−0.678496	−0.0251640	−0.0125820	−	0.999921i	$-0.504005\pi$
$727$	−0.678496	−0.0251640	−0.0125820	+	0.999921i	$0.504005\pi$
$728$	0	0
$729$	15.3344	0.567939
$730$	0	0
$731$	−22.1978	−0.821015
$732$	0	0
$733$	−29.0262	−1.07211	−0.536054	−	0.844184i	$-0.680086\pi$
$733$	−29.0262	−1.07211	−0.536054	+	0.844184i	$0.680086\pi$
$734$	0	0
$735$	42.3115	1.56068
$736$	0	0
$737$	−17.7463	−0.653693
$738$	0	0
$739$	25.5411	0.939543	0.469771	−	0.882788i	$-0.344336\pi$
$739$	25.5411	0.939543	0.469771	+	0.882788i	$0.344336\pi$
$740$	0	0
$741$	15.3635	0.564391
$742$	0	0
$743$	15.2865	0.560807	0.280404	−	0.959882i	$-0.409532\pi$
$743$	15.2865	0.560807	0.280404	+	0.959882i	$0.409532\pi$
$744$	0	0
$745$	88.9158	3.25762
$746$	0	0
$747$	−1.14794	−0.0420009
$748$	0	0
$749$	−23.6889	−0.865573
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	−28.7947	−1.04934
$754$	0	0
$755$	−85.7095	−3.11929
$756$	0	0
$757$	−50.9060	−1.85021	−0.925105	−	0.379712i	$-0.876023\pi$
$757$	−50.9060	−1.85021	−0.925105	+	0.379712i	$0.876023\pi$
$758$	0	0
$759$	−18.9333	−0.687237
$760$	0	0
$761$	39.0133	1.41423	0.707115	−	0.707098i	$-0.249996\pi$
$761$	39.0133	1.41423	0.707115	+	0.707098i	$0.249996\pi$
$762$	0	0
$763$	70.7046	2.55968
$764$	0	0
$765$	26.2746	0.949962
$766$	0	0
$767$	−65.4137	−2.36195
$768$	0	0
$769$	45.4979	1.64070	0.820348	−	0.571865i	$-0.193780\pi$
$769$	45.4979	1.64070	0.820348	+	0.571865i	$0.193780\pi$
$770$	0	0
$771$	−9.55190	−0.344003
$772$	0	0
$773$	−4.57106	−0.164410	−0.0822048	−	0.996615i	$-0.526196\pi$
$773$	−4.57106	−0.164410	−0.0822048	+	0.996615i	$0.526196\pi$
$774$	0	0
$775$	0.245213	0.00880832
$776$	0	0
$777$	−0.205901	−0.00738665
$778$	0	0
$779$	4.75363	0.170317
$780$	0	0
$781$	−24.1169	−0.862969
$782$	0	0
$783$	5.45241	0.194853
$784$	0	0
$785$	−49.6269	−1.77126
$786$	0	0
$787$	26.4115	0.941468	0.470734	−	0.882275i	$-0.343989\pi$
$787$	26.4115	0.941468	0.470734	+	0.882275i	$0.343989\pi$
$788$	0	0
$789$	−12.4136	−0.441935
$790$	0	0
$791$	5.64582	0.200742
$792$	0	0
$793$	14.2377	0.505596
$794$	0	0
$795$	7.62340	0.270374
$796$	0	0
$797$	13.3184	0.471760	0.235880	−	0.971782i	$-0.424203\pi$
$797$	13.3184	0.471760	0.235880	+	0.971782i	$0.424203\pi$
$798$	0	0
$799$	12.9833	0.459317
$800$	0	0
$801$	−0.440075	−0.0155493
$802$	0	0
$803$	11.4244	0.403157
$804$	0	0
$805$	−129.148	−4.55187
$806$	0	0
$807$	−3.06228	−0.107798
$808$	0	0
$809$	−37.1278	−1.30534	−0.652672	−	0.757640i	$-0.726352\pi$
$809$	−37.1278	−1.30534	−0.652672	+	0.757640i	$0.726352\pi$
$810$	0	0
$811$	19.2508	0.675989	0.337994	−	0.941148i	$-0.390252\pi$
$811$	19.2508	0.675989	0.337994	+	0.941148i	$0.390252\pi$
$812$	0	0
$813$	13.0717	0.458446
$814$	0	0
$815$	28.7716	1.00782
$816$	0	0
$817$	14.6678	0.513163
$818$	0	0
$819$	−49.7392	−1.73803
$820$	0	0
$821$	−5.04722	−0.176149	−0.0880746	−	0.996114i	$-0.528071\pi$
$821$	−5.04722	−0.176149	−0.0880746	+	0.996114i	$0.528071\pi$
$822$	0	0
$823$	−3.43300	−0.119667	−0.0598335	−	0.998208i	$-0.519057\pi$
$823$	−3.43300	−0.119667	−0.0598335	+	0.998208i	$0.519057\pi$
$824$	0	0
$825$	−22.9734	−0.799832
$826$	0	0
$827$	13.3732	0.465030	0.232515	−	0.972593i	$-0.425305\pi$
$827$	13.3732	0.465030	0.232515	+	0.972593i	$0.425305\pi$
$828$	0	0
$829$	−50.2788	−1.74626	−0.873128	−	0.487492i	$-0.837912\pi$
$829$	−50.2788	−1.74626	−0.873128	+	0.487492i	$0.837912\pi$
$830$	0	0
$831$	1.39873	0.0485213
$832$	0	0
$833$	37.8195	1.31037
$834$	0	0
$835$	−34.2867	−1.18654
$836$	0	0
$837$	0.129041	0.00446031
$838$	0	0
$839$	38.8844	1.34244	0.671219	−	0.741259i	$-0.265771\pi$
$839$	38.8844	1.34244	0.671219	+	0.741259i	$0.265771\pi$
$840$	0	0
$841$	−27.8687	−0.960989
$842$	0	0
$843$	−6.68632	−0.230289
$844$	0	0
$845$	−98.4092	−3.38538
$846$	0	0
$847$	−24.7100	−0.849045
$848$	0	0
$849$	26.6407	0.914308
$850$	0	0
$851$	0.377877	0.0129534
$852$	0	0
$853$	−11.8791	−0.406734	−0.203367	−	0.979103i	$-0.565188\pi$
$853$	−11.8791	−0.406734	−0.203367	+	0.979103i	$0.565188\pi$
$854$	0	0
$855$	−17.3617	−0.593759
$856$	0	0
$857$	17.1359	0.585351	0.292676	−	0.956212i	$-0.405454\pi$
$857$	17.1359	0.585351	0.292676	+	0.956212i	$0.405454\pi$
$858$	0	0
$859$	−5.98294	−0.204135	−0.102068	−	0.994777i	$-0.532546\pi$
$859$	−5.98294	−0.204135	−0.102068	+	0.994777i	$0.532546\pi$
$860$	0	0
$861$	8.78314	0.299328
$862$	0	0
$863$	−1.96387	−0.0668511	−0.0334255	−	0.999441i	$-0.510642\pi$
$863$	−1.96387	−0.0668511	−0.0334255	+	0.999441i	$0.510642\pi$
$864$	0	0
$865$	92.9107	3.15906
$866$	0	0
$867$	4.34561	0.147585
$868$	0	0
$869$	32.0857	1.08843
$870$	0	0
$871$	−48.8298	−1.65454
$872$	0	0
$873$	3.53360	0.119594
$874$	0	0
$875$	−76.2718	−2.57846
$876$	0	0
$877$	−19.5005	−0.658486	−0.329243	−	0.944245i	$-0.606794\pi$
$877$	−19.5005	−0.658486	−0.329243	+	0.944245i	$0.606794\pi$
$878$	0	0
$879$	−7.01406	−0.236578
$880$	0	0
$881$	−7.19109	−0.242274	−0.121137	−	0.992636i	$-0.538654\pi$
$881$	−7.19109	−0.242274	−0.121137	+	0.992636i	$0.538654\pi$
$882$	0	0
$883$	−58.2224	−1.95934	−0.979669	−	0.200620i	$-0.935704\pi$
$883$	−58.2224	−1.95934	−0.979669	+	0.200620i	$0.935704\pi$
$884$	0	0
$885$	−42.1878	−1.41813
$886$	0	0
$887$	−37.3505	−1.25411	−0.627053	−	0.778977i	$-0.715739\pi$
$887$	−37.3505	−1.25411	−0.627053	+	0.778977i	$0.715739\pi$
$888$	0	0
$889$	64.9499	2.17835
$890$	0	0
$891$	0.853566	0.0285955
$892$	0	0
$893$	−8.57911	−0.287089
$894$	0	0
$895$	−3.08026	−0.102962
$896$	0	0
$897$	−52.0961	−1.73944
$898$	0	0
$899$	0.0267744	0.000892976 0
$900$	0	0
$901$	6.81405	0.227009
$902$	0	0
$903$	27.1013	0.901875
$904$	0	0
$905$	64.6780	2.14997
$906$	0	0
$907$	−16.6227	−0.551947	−0.275973	−	0.961165i	$-0.589000\pi$
$907$	−16.6227	−0.551947	−0.275973	+	0.961165i	$0.589000\pi$
$908$	0	0
$909$	−15.8975	−0.527286
$910$	0	0
$911$	−20.7689	−0.688103	−0.344052	−	0.938951i	$-0.611799\pi$
$911$	−20.7689	−0.688103	−0.344052	+	0.938951i	$0.611799\pi$
$912$	0	0
$913$	1.35764	0.0449313
$914$	0	0
$915$	9.18244	0.303562
$916$	0	0
$917$	19.1889	0.633674
$918$	0	0
$919$	−42.1057	−1.38894	−0.694469	−	0.719523i	$-0.744361\pi$
$919$	−42.1057	−1.38894	−0.694469	+	0.719523i	$0.744361\pi$
$920$	0	0
$921$	−20.6395	−0.680095
$922$	0	0
$923$	−66.3588	−2.18423
$924$	0	0
$925$	0.458509	0.0150757
$926$	0	0
$927$	30.8755	1.01408
$928$	0	0
$929$	32.5127	1.06671	0.533354	−	0.845892i	$-0.320931\pi$
$929$	32.5127	1.06671	0.533354	+	0.845892i	$0.320931\pi$
$930$	0	0
$931$	−24.9903	−0.819025
$932$	0	0
$933$	−8.19317	−0.268232
$934$	0	0
$935$	−31.0744	−1.01624
$936$	0	0
$937$	−17.1587	−0.560551	−0.280275	−	0.959920i	$-0.590426\pi$
$937$	−17.1587	−0.560551	−0.280275	+	0.959920i	$0.590426\pi$
$938$	0	0
$939$	−5.85905	−0.191203
$940$	0	0
$941$	−21.7037	−0.707522	−0.353761	−	0.935336i	$-0.615097\pi$
$941$	−21.7037	−0.707522	−0.353761	+	0.935336i	$0.615097\pi$
$942$	0	0
$943$	−16.1191	−0.524911
$944$	0	0
$945$	−82.4651	−2.68259
$946$	0	0
$947$	11.9303	0.387684	0.193842	−	0.981033i	$-0.437905\pi$
$947$	11.9303	0.387684	0.193842	+	0.981033i	$0.437905\pi$
$948$	0	0
$949$	31.4348	1.02042
$950$	0	0
$951$	33.6929	1.09257
$952$	0	0
$953$	−37.0792	−1.20111	−0.600557	−	0.799582i	$-0.705055\pi$
$953$	−37.0792	−1.20111	−0.600557	+	0.799582i	$0.705055\pi$
$954$	0	0
$955$	−8.86407	−0.286835
$956$	0	0
$957$	−2.50843	−0.0810859
$958$	0	0
$959$	−46.5270	−1.50244
$960$	0	0
$961$	−30.9994	−0.999980
$962$	0	0
$963$	10.7986	0.347979
$964$	0	0
$965$	−25.4567	−0.819479
$966$	0	0
$967$	−19.7594	−0.635420	−0.317710	−	0.948188i	$-0.602914\pi$
$967$	−19.7594	−0.635420	−0.317710	+	0.948188i	$0.602914\pi$
$968$	0	0
$969$	8.85654	0.284513
$970$	0	0
$971$	−20.8939	−0.670517	−0.335259	−	0.942126i	$-0.608824\pi$
$971$	−20.8939	−0.670517	−0.335259	+	0.942126i	$0.608824\pi$
$972$	0	0
$973$	77.0238	2.46927
$974$	0	0
$975$	−63.2125	−2.02442
$976$	0	0
$977$	6.42485	0.205549	0.102775	−	0.994705i	$-0.467228\pi$
$977$	6.42485	0.205549	0.102775	+	0.994705i	$0.467228\pi$
$978$	0	0
$979$	0.520466	0.0166342
$980$	0	0
$981$	−32.2307	−1.02905
$982$	0	0
$983$	29.2517	0.932985	0.466493	−	0.884525i	$-0.345517\pi$
$983$	29.2517	0.932985	0.466493	+	0.884525i	$0.345517\pi$
$984$	0	0
$985$	25.1214	0.800435
$986$	0	0
$987$	−15.8513	−0.504554
$988$	0	0
$989$	−49.7373	−1.58155
$990$	0	0
$991$	31.1507	0.989536	0.494768	−	0.869025i	$-0.335253\pi$
$991$	31.1507	0.989536	0.494768	+	0.869025i	$0.335253\pi$
$992$	0	0
$993$	−23.1619	−0.735021
$994$	0	0
$995$	−30.0449	−0.952488
$996$	0	0
$997$	30.8451	0.976873	0.488436	−	0.872599i	$-0.337567\pi$
$997$	30.8451	0.976873	0.488436	+	0.872599i	$0.337567\pi$
$998$	0	0
$999$	0.241286	0.00763395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.17	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.17	✓	49	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)

\(f(q)\)	\(=\)	\(q-1.04405 q^{3} -3.83943 q^{5} +4.18991 q^{7} -1.90997 q^{9} +O(q^{10})\)	\(q-1.04405 q^{3} -3.83943 q^{5} +4.18991 q^{7} -1.90997 q^{9} +2.25887 q^{11} +6.21540 q^{13} +4.00855 q^{15} +3.58298 q^{17} -2.36756 q^{19} -4.37446 q^{21} +8.02816 q^{23} +9.74125 q^{25} +5.12623 q^{27} +1.06363 q^{29} +0.0251727 q^{31} -2.35837 q^{33} -16.0869 q^{35} +0.0470689 q^{37} -6.48916 q^{39} -2.00782 q^{41} -6.19535 q^{43} +7.33319 q^{45} +3.62361 q^{47} +10.5553 q^{49} -3.74080 q^{51} +1.90179 q^{53} -8.67278 q^{55} +2.47184 q^{57} -10.5245 q^{59} +2.29071 q^{61} -8.00258 q^{63} -23.8636 q^{65} -7.85627 q^{67} -8.38178 q^{69} -10.6765 q^{71} +5.05756 q^{73} -10.1703 q^{75} +9.46446 q^{77} +14.2043 q^{79} +0.377873 q^{81} +0.601026 q^{83} -13.7566 q^{85} -1.11048 q^{87} +0.230410 q^{89} +26.0419 q^{91} -0.0262815 q^{93} +9.09007 q^{95} -1.85008 q^{97} -4.31437 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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