LMFDB - Embedded newform 6008.2.a.d.1.10

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			1.10
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)

$f(q)$	$=$	$q-1.95337 q^{3} +3.19909 q^{5} +4.95256 q^{7} +0.815657 q^{9} +O(q^{10})$	$q-1.95337 q^{3} +3.19909 q^{5} +4.95256 q^{7} +0.815657 q^{9} -0.512980 q^{11} +6.09285 q^{13} -6.24901 q^{15} -5.31876 q^{17} +5.55689 q^{19} -9.67418 q^{21} -0.279596 q^{23} +5.23417 q^{25} +4.26683 q^{27} +0.455127 q^{29} -3.23878 q^{31} +1.00204 q^{33} +15.8437 q^{35} -8.61798 q^{37} -11.9016 q^{39} +1.45671 q^{41} +5.88874 q^{43} +2.60936 q^{45} +6.09428 q^{47} +17.5278 q^{49} +10.3895 q^{51} +1.06112 q^{53} -1.64107 q^{55} -10.8547 q^{57} +4.58507 q^{59} +0.389958 q^{61} +4.03959 q^{63} +19.4916 q^{65} -0.0619307 q^{67} +0.546154 q^{69} +3.23682 q^{71} +11.3647 q^{73} -10.2243 q^{75} -2.54056 q^{77} -3.20559 q^{79} -10.7817 q^{81} +1.53040 q^{83} -17.0152 q^{85} -0.889031 q^{87} +3.21282 q^{89} +30.1752 q^{91} +6.32653 q^{93} +17.7770 q^{95} -9.63886 q^{97} -0.418415 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.95337	−1.12778	−0.563890	−	0.825850i	$-0.690696\pi$
$3$	−1.95337	−1.12778	−0.563890	+	0.825850i	$0.690696\pi$
$4$	0	0
$5$	3.19909	1.43068	0.715338	−	0.698778i	$-0.246273\pi$
$5$	3.19909	1.43068	0.715338	+	0.698778i	$0.246273\pi$
$6$	0	0
$7$	4.95256	1.87189	0.935945	−	0.352145i	$-0.114548\pi$
$7$	4.95256	1.87189	0.935945	+	0.352145i	$0.114548\pi$
$8$	0	0
$9$	0.815657	0.271886
$10$	0	0
$11$	−0.512980	−0.154669	−0.0773346	−	0.997005i	$-0.524641\pi$
$11$	−0.512980	−0.154669	−0.0773346	+	0.997005i	$0.524641\pi$
$12$	0	0
$13$	6.09285	1.68985	0.844926	−	0.534883i	$-0.179644\pi$
$13$	6.09285	1.68985	0.844926	+	0.534883i	$0.179644\pi$
$14$	0	0
$15$	−6.24901	−1.61349
$16$	0	0
$17$	−5.31876	−1.28999	−0.644995	−	0.764187i	$-0.723140\pi$
$17$	−5.31876	−1.28999	−0.644995	+	0.764187i	$0.723140\pi$
$18$	0	0
$19$	5.55689	1.27484	0.637419	−	0.770517i	$-0.280002\pi$
$19$	5.55689	1.27484	0.637419	+	0.770517i	$0.280002\pi$
$20$	0	0
$21$	−9.67418	−2.11108
$22$	0	0
$23$	−0.279596	−0.0582997	−0.0291499	−	0.999575i	$-0.509280\pi$
$23$	−0.279596	−0.0582997	−0.0291499	+	0.999575i	$0.509280\pi$
$24$	0	0
$25$	5.23417	1.04683
$26$	0	0
$27$	4.26683	0.821152
$28$	0	0
$29$	0.455127	0.0845149	0.0422575	−	0.999107i	$-0.486545\pi$
$29$	0.455127	0.0845149	0.0422575	+	0.999107i	$0.486545\pi$
$30$	0	0
$31$	−3.23878	−0.581702	−0.290851	−	0.956768i	$-0.593938\pi$
$31$	−3.23878	−0.581702	−0.290851	+	0.956768i	$0.593938\pi$
$32$	0	0
$33$	1.00204	0.174433
$34$	0	0
$35$	15.8437	2.67807
$36$	0	0
$37$	−8.61798	−1.41679	−0.708393	−	0.705818i	$-0.750580\pi$
$37$	−8.61798	−1.41679	−0.708393	+	0.705818i	$0.750580\pi$
$38$	0	0
$39$	−11.9016	−1.90578
$40$	0	0
$41$	1.45671	0.227499	0.113750	−	0.993509i	$-0.463714\pi$
$41$	1.45671	0.227499	0.113750	+	0.993509i	$0.463714\pi$
$42$	0	0
$43$	5.88874	0.898025	0.449013	−	0.893525i	$-0.351776\pi$
$43$	5.88874	0.898025	0.449013	+	0.893525i	$0.351776\pi$
$44$	0	0
$45$	2.60936	0.388980
$46$	0	0
$47$	6.09428	0.888942	0.444471	−	0.895793i	$-0.353392\pi$
$47$	6.09428	0.888942	0.444471	+	0.895793i	$0.353392\pi$
$48$	0	0
$49$	17.5278	2.50397
$50$	0	0
$51$	10.3895	1.45482
$52$	0	0
$53$	1.06112	0.145756	0.0728780	−	0.997341i	$-0.476782\pi$
$53$	1.06112	0.145756	0.0728780	+	0.997341i	$0.476782\pi$
$54$	0	0
$55$	−1.64107	−0.221281
$56$	0	0
$57$	−10.8547	−1.43774
$58$	0	0
$59$	4.58507	0.596925	0.298462	−	0.954421i	$-0.403526\pi$
$59$	4.58507	0.596925	0.298462	+	0.954421i	$0.403526\pi$
$60$	0	0
$61$	0.389958	0.0499291	0.0249645	−	0.999688i	$-0.492053\pi$
$61$	0.389958	0.0499291	0.0249645	+	0.999688i	$0.492053\pi$
$62$	0	0
$63$	4.03959	0.508940
$64$	0	0
$65$	19.4916	2.41763
$66$	0	0
$67$	−0.0619307	−0.00756604	−0.00378302	−	0.999993i	$-0.501204\pi$
$67$	−0.0619307	−0.00756604	−0.00378302	+	0.999993i	$0.501204\pi$
$68$	0	0
$69$	0.546154	0.0657492
$70$	0	0
$71$	3.23682	0.384140	0.192070	−	0.981381i	$-0.438480\pi$
$71$	3.23682	0.384140	0.192070	+	0.981381i	$0.438480\pi$
$72$	0	0
$73$	11.3647	1.33014	0.665069	−	0.746782i	$-0.268402\pi$
$73$	11.3647	1.33014	0.665069	+	0.746782i	$0.268402\pi$
$74$	0	0
$75$	−10.2243	−1.18060
$76$	0	0
$77$	−2.54056	−0.289524
$78$	0	0
$79$	−3.20559	−0.360657	−0.180328	−	0.983606i	$-0.557716\pi$
$79$	−3.20559	−0.360657	−0.180328	+	0.983606i	$0.557716\pi$
$80$	0	0
$81$	−10.7817	−1.19796
$82$	0	0
$83$	1.53040	0.167983	0.0839914	−	0.996466i	$-0.473233\pi$
$83$	1.53040	0.167983	0.0839914	+	0.996466i	$0.473233\pi$
$84$	0	0
$85$	−17.0152	−1.84556
$86$	0	0
$87$	−0.889031	−0.0953142
$88$	0	0
$89$	3.21282	0.340558	0.170279	−	0.985396i	$-0.445533\pi$
$89$	3.21282	0.340558	0.170279	+	0.985396i	$0.445533\pi$
$90$	0	0
$91$	30.1752	3.16322
$92$	0	0
$93$	6.32653	0.656031
$94$	0	0
$95$	17.7770	1.82388
$96$	0	0
$97$	−9.63886	−0.978678	−0.489339	−	0.872094i	$-0.662762\pi$
$97$	−9.63886	−0.978678	−0.489339	+	0.872094i	$0.662762\pi$
$98$	0	0
$99$	−0.418415	−0.0420523
$100$	0	0
$101$	−18.3890	−1.82977	−0.914885	−	0.403714i	$-0.867719\pi$
$101$	−18.3890	−1.82977	−0.914885	+	0.403714i	$0.867719\pi$
$102$	0	0
$103$	−4.43734	−0.437224	−0.218612	−	0.975812i	$-0.570153\pi$
$103$	−4.43734	−0.437224	−0.218612	+	0.975812i	$0.570153\pi$
$104$	0	0
$105$	−30.9486	−3.02027
$106$	0	0
$107$	6.88946	0.666030	0.333015	−	0.942922i	$-0.391934\pi$
$107$	6.88946	0.666030	0.333015	+	0.942922i	$0.391934\pi$
$108$	0	0
$109$	−0.196589	−0.0188298	−0.00941492	−	0.999956i	$-0.502997\pi$
$109$	−0.196589	−0.0188298	−0.00941492	+	0.999956i	$0.502997\pi$
$110$	0	0
$111$	16.8341	1.59782
$112$	0	0
$113$	−2.36701	−0.222669	−0.111335	−	0.993783i	$-0.535513\pi$
$113$	−2.36701	−0.222669	−0.111335	+	0.993783i	$0.535513\pi$
$114$	0	0
$115$	−0.894451	−0.0834080
$116$	0	0
$117$	4.96968	0.459447
$118$	0	0
$119$	−26.3415	−2.41472
$120$	0	0
$121$	−10.7369	−0.976077
$122$	0	0
$123$	−2.84549	−0.256569
$124$	0	0
$125$	0.749141	0.0670052
$126$	0	0
$127$	19.8312	1.75974	0.879869	−	0.475217i	$-0.157630\pi$
$127$	19.8312	1.75974	0.879869	+	0.475217i	$0.157630\pi$
$128$	0	0
$129$	−11.5029	−1.01277
$130$	0	0
$131$	−0.755280	−0.0659891	−0.0329946	−	0.999456i	$-0.510504\pi$
$131$	−0.755280	−0.0659891	−0.0329946	+	0.999456i	$0.510504\pi$
$132$	0	0
$133$	27.5208	2.38636
$134$	0	0
$135$	13.6500	1.17480
$136$	0	0
$137$	−12.8950	−1.10170	−0.550848	−	0.834605i	$-0.685696\pi$
$137$	−12.8950	−1.10170	−0.550848	+	0.834605i	$0.685696\pi$
$138$	0	0
$139$	8.64665	0.733399	0.366699	−	0.930339i	$-0.380488\pi$
$139$	8.64665	0.733399	0.366699	+	0.930339i	$0.380488\pi$
$140$	0	0
$141$	−11.9044	−1.00253
$142$	0	0
$143$	−3.12551	−0.261368
$144$	0	0
$145$	1.45599	0.120913
$146$	0	0
$147$	−34.2383	−2.82393
$148$	0	0
$149$	−4.86828	−0.398825	−0.199413	−	0.979916i	$-0.563903\pi$
$149$	−4.86828	−0.398825	−0.199413	+	0.979916i	$0.563903\pi$
$150$	0	0
$151$	−3.50996	−0.285637	−0.142818	−	0.989749i	$-0.545616\pi$
$151$	−3.50996	−0.285637	−0.142818	+	0.989749i	$0.545616\pi$
$152$	0	0
$153$	−4.33829	−0.350730
$154$	0	0
$155$	−10.3611	−0.832227
$156$	0	0
$157$	−0.384398	−0.0306783	−0.0153391	−	0.999882i	$-0.504883\pi$
$157$	−0.384398	−0.0306783	−0.0153391	+	0.999882i	$0.504883\pi$
$158$	0	0
$159$	−2.07276	−0.164380
$160$	0	0
$161$	−1.38471	−0.109131
$162$	0	0
$163$	−7.14323	−0.559501	−0.279750	−	0.960073i	$-0.590252\pi$
$163$	−7.14323	−0.559501	−0.279750	+	0.960073i	$0.590252\pi$
$164$	0	0
$165$	3.20561	0.249557
$166$	0	0
$167$	−17.6679	−1.36718	−0.683590	−	0.729866i	$-0.739583\pi$
$167$	−17.6679	−1.36718	−0.683590	+	0.729866i	$0.739583\pi$
$168$	0	0
$169$	24.1228	1.85560
$170$	0	0
$171$	4.53252	0.346610
$172$	0	0
$173$	−5.78200	−0.439598	−0.219799	−	0.975545i	$-0.570540\pi$
$173$	−5.78200	−0.439598	−0.219799	+	0.975545i	$0.570540\pi$
$174$	0	0
$175$	25.9225	1.95956
$176$	0	0
$177$	−8.95634	−0.673199
$178$	0	0
$179$	19.0620	1.42476	0.712381	−	0.701793i	$-0.247617\pi$
$179$	19.0620	1.42476	0.712381	+	0.701793i	$0.247617\pi$
$180$	0	0
$181$	25.1650	1.87050	0.935252	−	0.353984i	$-0.115173\pi$
$181$	25.1650	1.87050	0.935252	+	0.353984i	$0.115173\pi$
$182$	0	0
$183$	−0.761733	−0.0563090
$184$	0	0
$185$	−27.5697	−2.02696
$186$	0	0
$187$	2.72842	0.199522
$188$	0	0
$189$	21.1317	1.53711
$190$	0	0
$191$	−9.30606	−0.673363	−0.336681	−	0.941619i	$-0.609305\pi$
$191$	−9.30606	−0.673363	−0.336681	+	0.941619i	$0.609305\pi$
$192$	0	0
$193$	−2.17563	−0.156605	−0.0783027	−	0.996930i	$-0.524950\pi$
$193$	−2.17563	−0.156605	−0.0783027	+	0.996930i	$0.524950\pi$
$194$	0	0
$195$	−38.0743	−2.72656
$196$	0	0
$197$	−12.8572	−0.916035	−0.458017	−	0.888943i	$-0.651440\pi$
$197$	−12.8572	−0.916035	−0.458017	+	0.888943i	$0.651440\pi$
$198$	0	0
$199$	−24.5970	−1.74364	−0.871818	−	0.489831i	$-0.837059\pi$
$199$	−24.5970	−1.74364	−0.871818	+	0.489831i	$0.837059\pi$
$200$	0	0
$201$	0.120974	0.00853282
$202$	0	0
$203$	2.25404	0.158203
$204$	0	0
$205$	4.66013	0.325478
$206$	0	0
$207$	−0.228054	−0.0158509
$208$	0	0
$209$	−2.85057	−0.197178
$210$	0	0
$211$	21.0964	1.45234	0.726168	−	0.687517i	$-0.241299\pi$
$211$	21.0964	1.45234	0.726168	+	0.687517i	$0.241299\pi$
$212$	0	0
$213$	−6.32270	−0.433225
$214$	0	0
$215$	18.8386	1.28478
$216$	0	0
$217$	−16.0402	−1.08888
$218$	0	0
$219$	−22.1995	−1.50010
$220$	0	0
$221$	−32.4064	−2.17989
$222$	0	0
$223$	−3.19443	−0.213915	−0.106957	−	0.994264i	$-0.534111\pi$
$223$	−3.19443	−0.213915	−0.106957	+	0.994264i	$0.534111\pi$
$224$	0	0
$225$	4.26929	0.284619
$226$	0	0
$227$	−24.2000	−1.60621	−0.803104	−	0.595839i	$-0.796820\pi$
$227$	−24.2000	−1.60621	−0.803104	+	0.595839i	$0.796820\pi$
$228$	0	0
$229$	6.90507	0.456300	0.228150	−	0.973626i	$-0.426732\pi$
$229$	6.90507	0.456300	0.228150	+	0.973626i	$0.426732\pi$
$230$	0	0
$231$	4.96266	0.326519
$232$	0	0
$233$	−20.8027	−1.36283	−0.681414	−	0.731899i	$-0.738634\pi$
$233$	−20.8027	−1.36283	−0.681414	+	0.731899i	$0.738634\pi$
$234$	0	0
$235$	19.4962	1.27179
$236$	0	0
$237$	6.26170	0.406741
$238$	0	0
$239$	10.6475	0.688728	0.344364	−	0.938836i	$-0.388095\pi$
$239$	10.6475	0.688728	0.344364	+	0.938836i	$0.388095\pi$
$240$	0	0
$241$	−19.2960	−1.24296	−0.621481	−	0.783429i	$-0.713469\pi$
$241$	−19.2960	−1.24296	−0.621481	+	0.783429i	$0.713469\pi$
$242$	0	0
$243$	8.26011	0.529886
$244$	0	0
$245$	56.0731	3.58238
$246$	0	0
$247$	33.8573	2.15429
$248$	0	0
$249$	−2.98943	−0.189447
$250$	0	0
$251$	−0.855934	−0.0540261	−0.0270131	−	0.999635i	$-0.508600\pi$
$251$	−0.855934	−0.0540261	−0.0270131	+	0.999635i	$0.508600\pi$
$252$	0	0
$253$	0.143427	0.00901717
$254$	0	0
$255$	33.2370	2.08138
$256$	0	0
$257$	15.1537	0.945262	0.472631	−	0.881261i	$-0.343304\pi$
$257$	15.1537	0.945262	0.472631	+	0.881261i	$0.343304\pi$
$258$	0	0
$259$	−42.6810	−2.65207
$260$	0	0
$261$	0.371227	0.0229784
$262$	0	0
$263$	11.4905	0.708535	0.354267	−	0.935144i	$-0.384730\pi$
$263$	11.4905	0.708535	0.354267	+	0.935144i	$0.384730\pi$
$264$	0	0
$265$	3.39461	0.208530
$266$	0	0
$267$	−6.27583	−0.384074
$268$	0	0
$269$	−32.7096	−1.99434	−0.997169	−	0.0751918i	$-0.976043\pi$
$269$	−32.7096	−1.99434	−0.997169	+	0.0751918i	$0.976043\pi$
$270$	0	0
$271$	25.9497	1.57633	0.788166	−	0.615462i	$-0.211031\pi$
$271$	25.9497	1.57633	0.788166	+	0.615462i	$0.211031\pi$
$272$	0	0
$273$	−58.9433	−3.56741
$274$	0	0
$275$	−2.68502	−0.161913
$276$	0	0
$277$	20.3198	1.22090	0.610449	−	0.792056i	$-0.290989\pi$
$277$	20.3198	1.22090	0.610449	+	0.792056i	$0.290989\pi$
$278$	0	0
$279$	−2.64173	−0.158156
$280$	0	0
$281$	14.8857	0.888008	0.444004	−	0.896025i	$-0.353558\pi$
$281$	14.8857	0.888008	0.444004	+	0.896025i	$0.353558\pi$
$282$	0	0
$283$	−24.2031	−1.43872	−0.719362	−	0.694635i	$-0.755566\pi$
$283$	−24.2031	−1.43872	−0.719362	+	0.694635i	$0.755566\pi$
$284$	0	0
$285$	−34.7251	−2.05694
$286$	0	0
$287$	7.21442	0.425853
$288$	0	0
$289$	11.2892	0.664073
$290$	0	0
$291$	18.8283	1.10373
$292$	0	0
$293$	4.01433	0.234520	0.117260	−	0.993101i	$-0.462589\pi$
$293$	4.01433	0.234520	0.117260	+	0.993101i	$0.462589\pi$
$294$	0	0
$295$	14.6680	0.854006
$296$	0	0
$297$	−2.18880	−0.127007
$298$	0	0
$299$	−1.70353	−0.0985179
$300$	0	0
$301$	29.1643	1.68100
$302$	0	0
$303$	35.9205	2.06358
$304$	0	0
$305$	1.24751	0.0714323
$306$	0	0
$307$	−14.6908	−0.838450	−0.419225	−	0.907882i	$-0.637698\pi$
$307$	−14.6908	−0.838450	−0.419225	+	0.907882i	$0.637698\pi$
$308$	0	0
$309$	8.66777	0.493092
$310$	0	0
$311$	13.8839	0.787283	0.393642	−	0.919264i	$-0.371215\pi$
$311$	13.8839	0.787283	0.393642	+	0.919264i	$0.371215\pi$
$312$	0	0
$313$	18.9665	1.07205	0.536026	−	0.844201i	$-0.319925\pi$
$313$	18.9665	1.07205	0.536026	+	0.844201i	$0.319925\pi$
$314$	0	0
$315$	12.9230	0.728129
$316$	0	0
$317$	−7.93223	−0.445518	−0.222759	−	0.974874i	$-0.571506\pi$
$317$	−7.93223	−0.445518	−0.222759	+	0.974874i	$0.571506\pi$
$318$	0	0
$319$	−0.233471	−0.0130719
$320$	0	0
$321$	−13.4577	−0.751134
$322$	0	0
$323$	−29.5558	−1.64453
$324$	0	0
$325$	31.8910	1.76900
$326$	0	0
$327$	0.384012	0.0212359
$328$	0	0
$329$	30.1823	1.66400
$330$	0	0
$331$	3.00396	0.165112	0.0825562	−	0.996586i	$-0.473692\pi$
$331$	3.00396	0.165112	0.0825562	+	0.996586i	$0.473692\pi$
$332$	0	0
$333$	−7.02931	−0.385204
$334$	0	0
$335$	−0.198122	−0.0108246
$336$	0	0
$337$	6.25964	0.340984	0.170492	−	0.985359i	$-0.445464\pi$
$337$	6.25964	0.340984	0.170492	+	0.985359i	$0.445464\pi$
$338$	0	0
$339$	4.62364	0.251122
$340$	0	0
$341$	1.66143	0.0899713
$342$	0	0
$343$	52.1396	2.81528
$344$	0	0
$345$	1.74719	0.0940658
$346$	0	0
$347$	24.5911	1.32012	0.660061	−	0.751212i	$-0.270531\pi$
$347$	24.5911	1.32012	0.660061	+	0.751212i	$0.270531\pi$
$348$	0	0
$349$	−20.5027	−1.09748	−0.548741	−	0.835992i	$-0.684893\pi$
$349$	−20.5027	−1.09748	−0.548741	+	0.835992i	$0.684893\pi$
$350$	0	0
$351$	25.9972	1.38763
$352$	0	0
$353$	29.1018	1.54893	0.774467	−	0.632614i	$-0.218018\pi$
$353$	29.1018	1.54893	0.774467	+	0.632614i	$0.218018\pi$
$354$	0	0
$355$	10.3549	0.549579
$356$	0	0
$357$	51.4547	2.72327
$358$	0	0
$359$	−3.38814	−0.178819	−0.0894095	−	0.995995i	$-0.528498\pi$
$359$	−3.38814	−0.178819	−0.0894095	+	0.995995i	$0.528498\pi$
$360$	0	0
$361$	11.8791	0.625214
$362$	0	0
$363$	20.9731	1.10080
$364$	0	0
$365$	36.3567	1.90300
$366$	0	0
$367$	−26.2945	−1.37256	−0.686280	−	0.727337i	$-0.740758\pi$
$367$	−26.2945	−1.37256	−0.686280	+	0.727337i	$0.740758\pi$
$368$	0	0
$369$	1.18817	0.0618537
$370$	0	0
$371$	5.25525	0.272839
$372$	0	0
$373$	−19.8878	−1.02975	−0.514876	−	0.857265i	$-0.672162\pi$
$373$	−19.8878	−1.02975	−0.514876	+	0.857265i	$0.672162\pi$
$374$	0	0
$375$	−1.46335	−0.0755671
$376$	0	0
$377$	2.77302	0.142818
$378$	0	0
$379$	4.68702	0.240756	0.120378	−	0.992728i	$-0.461589\pi$
$379$	4.68702	0.240756	0.120378	+	0.992728i	$0.461589\pi$
$380$	0	0
$381$	−38.7377	−1.98459
$382$	0	0
$383$	−4.93512	−0.252173	−0.126086	−	0.992019i	$-0.540242\pi$
$383$	−4.93512	−0.252173	−0.126086	+	0.992019i	$0.540242\pi$
$384$	0	0
$385$	−8.12748	−0.414215
$386$	0	0
$387$	4.80319	0.244160
$388$	0	0
$389$	21.7973	1.10517	0.552584	−	0.833457i	$-0.313642\pi$
$389$	21.7973	1.10517	0.552584	+	0.833457i	$0.313642\pi$
$390$	0	0
$391$	1.48710	0.0752060
$392$	0	0
$393$	1.47534	0.0744211
$394$	0	0
$395$	−10.2550	−0.515983
$396$	0	0
$397$	−15.2271	−0.764226	−0.382113	−	0.924116i	$-0.624804\pi$
$397$	−15.2271	−0.764226	−0.382113	+	0.924116i	$0.624804\pi$
$398$	0	0
$399$	−53.7584	−2.69129
$400$	0	0
$401$	2.15376	0.107554	0.0537769	−	0.998553i	$-0.482874\pi$
$401$	2.15376	0.107554	0.0537769	+	0.998553i	$0.482874\pi$
$402$	0	0
$403$	−19.7334	−0.982990
$404$	0	0
$405$	−34.4915	−1.71390
$406$	0	0
$407$	4.42085	0.219133
$408$	0	0
$409$	9.35748	0.462697	0.231349	−	0.972871i	$-0.425686\pi$
$409$	9.35748	0.462697	0.231349	+	0.972871i	$0.425686\pi$
$410$	0	0
$411$	25.1888	1.24247
$412$	0	0
$413$	22.7078	1.11738
$414$	0	0
$415$	4.89587	0.240329
$416$	0	0
$417$	−16.8901	−0.827112
$418$	0	0
$419$	23.4691	1.14654	0.573270	−	0.819366i	$-0.305675\pi$
$419$	23.4691	1.14654	0.573270	+	0.819366i	$0.305675\pi$
$420$	0	0
$421$	−30.9731	−1.50953	−0.754767	−	0.655993i	$-0.772250\pi$
$421$	−30.9731	−1.50953	−0.754767	+	0.655993i	$0.772250\pi$
$422$	0	0
$423$	4.97084	0.241691
$424$	0	0
$425$	−27.8393	−1.35041
$426$	0	0
$427$	1.93129	0.0934617
$428$	0	0
$429$	6.10528	0.294765
$430$	0	0
$431$	19.9779	0.962300	0.481150	−	0.876638i	$-0.340219\pi$
$431$	19.9779	0.962300	0.481150	+	0.876638i	$0.340219\pi$
$432$	0	0
$433$	29.3494	1.41044	0.705222	−	0.708987i	$-0.250847\pi$
$433$	29.3494	1.41044	0.705222	+	0.708987i	$0.250847\pi$
$434$	0	0
$435$	−2.84409	−0.136364
$436$	0	0
$437$	−1.55368	−0.0743227
$438$	0	0
$439$	31.9877	1.52669	0.763345	−	0.645991i	$-0.223556\pi$
$439$	31.9877	1.52669	0.763345	+	0.645991i	$0.223556\pi$
$440$	0	0
$441$	14.2967	0.680795
$442$	0	0
$443$	−19.7943	−0.940457	−0.470229	−	0.882545i	$-0.655829\pi$
$443$	−19.7943	−0.940457	−0.470229	+	0.882545i	$0.655829\pi$
$444$	0	0
$445$	10.2781	0.487229
$446$	0	0
$447$	9.50956	0.449787
$448$	0	0
$449$	−25.5416	−1.20538	−0.602691	−	0.797974i	$-0.705905\pi$
$449$	−25.5416	−1.20538	−0.602691	+	0.797974i	$0.705905\pi$
$450$	0	0
$451$	−0.747260	−0.0351871
$452$	0	0
$453$	6.85626	0.322135
$454$	0	0
$455$	96.5331	4.52554
$456$	0	0
$457$	22.1338	1.03537	0.517687	−	0.855570i	$-0.326793\pi$
$457$	22.1338	1.03537	0.517687	+	0.855570i	$0.326793\pi$
$458$	0	0
$459$	−22.6943	−1.05928
$460$	0	0
$461$	−5.15545	−0.240113	−0.120057	−	0.992767i	$-0.538308\pi$
$461$	−5.15545	−0.240113	−0.120057	+	0.992767i	$0.538308\pi$
$462$	0	0
$463$	24.1816	1.12381	0.561907	−	0.827200i	$-0.310068\pi$
$463$	24.1816	1.12381	0.561907	+	0.827200i	$0.310068\pi$
$464$	0	0
$465$	20.2391	0.938568
$466$	0	0
$467$	−15.9863	−0.739757	−0.369879	−	0.929080i	$-0.620601\pi$
$467$	−15.9863	−0.739757	−0.369879	+	0.929080i	$0.620601\pi$
$468$	0	0
$469$	−0.306715	−0.0141628
$470$	0	0
$471$	0.750871	0.0345983
$472$	0	0
$473$	−3.02081	−0.138897
$474$	0	0
$475$	29.0857	1.33455
$476$	0	0
$477$	0.865509	0.0396289
$478$	0	0
$479$	8.34411	0.381252	0.190626	−	0.981663i	$-0.438948\pi$
$479$	8.34411	0.381252	0.190626	+	0.981663i	$0.438948\pi$
$480$	0	0
$481$	−52.5081	−2.39416
$482$	0	0
$483$	2.70486	0.123075
$484$	0	0
$485$	−30.8356	−1.40017
$486$	0	0
$487$	30.4939	1.38181	0.690906	−	0.722945i	$-0.257212\pi$
$487$	30.4939	1.38181	0.690906	+	0.722945i	$0.257212\pi$
$488$	0	0
$489$	13.9534	0.630993
$490$	0	0
$491$	−24.0578	−1.08571	−0.542856	−	0.839826i	$-0.682657\pi$
$491$	−24.0578	−1.08571	−0.542856	+	0.839826i	$0.682657\pi$
$492$	0	0
$493$	−2.42071	−0.109023
$494$	0	0
$495$	−1.33855	−0.0601633
$496$	0	0
$497$	16.0305	0.719067
$498$	0	0
$499$	2.43347	0.108937	0.0544686	−	0.998515i	$-0.482654\pi$
$499$	2.43347	0.108937	0.0544686	+	0.998515i	$0.482654\pi$
$500$	0	0
$501$	34.5119	1.54188
$502$	0	0
$503$	−11.3936	−0.508017	−0.254008	−	0.967202i	$-0.581749\pi$
$503$	−11.3936	−0.508017	−0.254008	+	0.967202i	$0.581749\pi$
$504$	0	0
$505$	−58.8279	−2.61781
$506$	0	0
$507$	−47.1208	−2.09271
$508$	0	0
$509$	−29.4262	−1.30429	−0.652146	−	0.758093i	$-0.726131\pi$
$509$	−29.4262	−1.30429	−0.652146	+	0.758093i	$0.726131\pi$
$510$	0	0
$511$	56.2843	2.48987
$512$	0	0
$513$	23.7103	1.04684
$514$	0	0
$515$	−14.1955	−0.625527
$516$	0	0
$517$	−3.12624	−0.137492
$518$	0	0
$519$	11.2944	0.495769
$520$	0	0
$521$	42.1299	1.84575	0.922873	−	0.385104i	$-0.125835\pi$
$521$	42.1299	1.84575	0.922873	+	0.385104i	$0.125835\pi$
$522$	0	0
$523$	−9.49955	−0.415386	−0.207693	−	0.978194i	$-0.566596\pi$
$523$	−9.49955	−0.415386	−0.207693	+	0.978194i	$0.566596\pi$
$524$	0	0
$525$	−50.6363	−2.20995
$526$	0	0
$527$	17.2263	0.750389
$528$	0	0
$529$	−22.9218	−0.996601
$530$	0	0
$531$	3.73984	0.162295
$532$	0	0
$533$	8.87549	0.384440
$534$	0	0
$535$	22.0400	0.952873
$536$	0	0
$537$	−37.2352	−1.60682
$538$	0	0
$539$	−8.99141	−0.387288
$540$	0	0
$541$	37.0282	1.59197	0.795983	−	0.605318i	$-0.206954\pi$
$541$	37.0282	1.59197	0.795983	+	0.605318i	$0.206954\pi$
$542$	0	0
$543$	−49.1567	−2.10951
$544$	0	0
$545$	−0.628907	−0.0269394
$546$	0	0
$547$	36.3855	1.55573	0.777866	−	0.628430i	$-0.216302\pi$
$547$	36.3855	1.55573	0.777866	+	0.628430i	$0.216302\pi$
$548$	0	0
$549$	0.318072	0.0135750
$550$	0	0
$551$	2.52909	0.107743
$552$	0	0
$553$	−15.8759	−0.675110
$554$	0	0
$555$	53.8538	2.28597
$556$	0	0
$557$	−15.3374	−0.649868	−0.324934	−	0.945737i	$-0.605342\pi$
$557$	−15.3374	−0.649868	−0.324934	+	0.945737i	$0.605342\pi$
$558$	0	0
$559$	35.8792	1.51753
$560$	0	0
$561$	−5.32961	−0.225016
$562$	0	0
$563$	−20.5679	−0.866834	−0.433417	−	0.901193i	$-0.642692\pi$
$563$	−20.5679	−0.866834	−0.433417	+	0.901193i	$0.642692\pi$
$564$	0	0
$565$	−7.57226	−0.318568
$566$	0	0
$567$	−53.3969	−2.24246
$568$	0	0
$569$	5.81326	0.243704	0.121852	−	0.992548i	$-0.461117\pi$
$569$	5.81326	0.243704	0.121852	+	0.992548i	$0.461117\pi$
$570$	0	0
$571$	6.65098	0.278335	0.139167	−	0.990269i	$-0.455557\pi$
$571$	6.65098	0.278335	0.139167	+	0.990269i	$0.455557\pi$
$572$	0	0
$573$	18.1782	0.759404
$574$	0	0
$575$	−1.46345	−0.0610302
$576$	0	0
$577$	−38.4943	−1.60254	−0.801270	−	0.598303i	$-0.795842\pi$
$577$	−38.4943	−1.60254	−0.801270	+	0.598303i	$0.795842\pi$
$578$	0	0
$579$	4.24981	0.176616
$580$	0	0
$581$	7.57937	0.314445
$582$	0	0
$583$	−0.544332	−0.0225439
$584$	0	0
$585$	15.8984	0.657319
$586$	0	0
$587$	−16.8684	−0.696234	−0.348117	−	0.937451i	$-0.613179\pi$
$587$	−16.8684	−0.696234	−0.348117	+	0.937451i	$0.613179\pi$
$588$	0	0
$589$	−17.9975	−0.741576
$590$	0	0
$591$	25.1148	1.03308
$592$	0	0
$593$	46.4415	1.90712	0.953561	−	0.301199i	$-0.0973869\pi$
$593$	46.4415	1.90712	0.953561	+	0.301199i	$0.0973869\pi$
$594$	0	0
$595$	−84.2687	−3.45468
$596$	0	0
$597$	48.0471	1.96644
$598$	0	0
$599$	−33.8593	−1.38345	−0.691726	−	0.722160i	$-0.743149\pi$
$599$	−33.8593	−1.38345	−0.691726	+	0.722160i	$0.743149\pi$
$600$	0	0
$601$	32.6533	1.33196	0.665979	−	0.745971i	$-0.268014\pi$
$601$	32.6533	1.33196	0.665979	+	0.745971i	$0.268014\pi$
$602$	0	0
$603$	−0.0505142	−0.00205710
$604$	0	0
$605$	−34.3481	−1.39645
$606$	0	0
$607$	2.68993	0.109181	0.0545904	−	0.998509i	$-0.482615\pi$
$607$	2.68993	0.109181	0.0545904	+	0.998509i	$0.482615\pi$
$608$	0	0
$609$	−4.40298	−0.178418
$610$	0	0
$611$	37.1316	1.50218
$612$	0	0
$613$	18.8339	0.760695	0.380348	−	0.924844i	$-0.375804\pi$
$613$	18.8339	0.760695	0.380348	+	0.924844i	$0.375804\pi$
$614$	0	0
$615$	−9.10296	−0.367067
$616$	0	0
$617$	25.7622	1.03715	0.518573	−	0.855033i	$-0.326463\pi$
$617$	25.7622	1.03715	0.518573	+	0.855033i	$0.326463\pi$
$618$	0	0
$619$	38.1743	1.53435	0.767177	−	0.641435i	$-0.221661\pi$
$619$	38.1743	1.53435	0.767177	+	0.641435i	$0.221661\pi$
$620$	0	0
$621$	−1.19299	−0.0478729
$622$	0	0
$623$	15.9117	0.637488
$624$	0	0
$625$	−23.7743	−0.950972
$626$	0	0
$627$	5.56823	0.222373
$628$	0	0
$629$	45.8370	1.82764
$630$	0	0
$631$	−18.5782	−0.739585	−0.369792	−	0.929114i	$-0.620571\pi$
$631$	−18.5782	−0.739585	−0.369792	+	0.929114i	$0.620571\pi$
$632$	0	0
$633$	−41.2091	−1.63791
$634$	0	0
$635$	63.4419	2.51761
$636$	0	0
$637$	106.794	4.23135
$638$	0	0
$639$	2.64013	0.104442
$640$	0	0
$641$	42.7336	1.68788	0.843938	−	0.536441i	$-0.180232\pi$
$641$	42.7336	1.68788	0.843938	+	0.536441i	$0.180232\pi$
$642$	0	0
$643$	14.6817	0.578989	0.289495	−	0.957180i	$-0.406513\pi$
$643$	14.6817	0.578989	0.289495	+	0.957180i	$0.406513\pi$
$644$	0	0
$645$	−36.7988	−1.44895
$646$	0	0
$647$	−25.9847	−1.02156	−0.510782	−	0.859710i	$-0.670644\pi$
$647$	−25.9847	−1.02156	−0.510782	+	0.859710i	$0.670644\pi$
$648$	0	0
$649$	−2.35205	−0.0923259
$650$	0	0
$651$	31.3325	1.22802
$652$	0	0
$653$	−2.99832	−0.117333	−0.0586666	−	0.998278i	$-0.518685\pi$
$653$	−2.99832	−0.117333	−0.0586666	+	0.998278i	$0.518685\pi$
$654$	0	0
$655$	−2.41621	−0.0944091
$656$	0	0
$657$	9.26970	0.361645
$658$	0	0
$659$	40.7330	1.58673	0.793366	−	0.608745i	$-0.208327\pi$
$659$	40.7330	1.58673	0.793366	+	0.608745i	$0.208327\pi$
$660$	0	0
$661$	6.80526	0.264694	0.132347	−	0.991203i	$-0.457749\pi$
$661$	6.80526	0.264694	0.132347	+	0.991203i	$0.457749\pi$
$662$	0	0
$663$	63.3018	2.45844
$664$	0	0
$665$	88.0416	3.41411
$666$	0	0
$667$	−0.127251	−0.00492720
$668$	0	0
$669$	6.23991	0.241249
$670$	0	0
$671$	−0.200041	−0.00772249
$672$	0	0
$673$	−35.9195	−1.38459	−0.692297	−	0.721612i	$-0.743401\pi$
$673$	−35.9195	−1.38459	−0.692297	+	0.721612i	$0.743401\pi$
$674$	0	0
$675$	22.3333	0.859611
$676$	0	0
$677$	16.3123	0.626933	0.313466	−	0.949599i	$-0.398510\pi$
$677$	16.3123	0.626933	0.313466	+	0.949599i	$0.398510\pi$
$678$	0	0
$679$	−47.7370	−1.83198
$680$	0	0
$681$	47.2715	1.81145
$682$	0	0
$683$	−43.4524	−1.66266	−0.831330	−	0.555779i	$-0.812420\pi$
$683$	−43.4524	−1.66266	−0.831330	+	0.555779i	$0.812420\pi$
$684$	0	0
$685$	−41.2524	−1.57617
$686$	0	0
$687$	−13.4882	−0.514605
$688$	0	0
$689$	6.46524	0.246306
$690$	0	0
$691$	−39.4013	−1.49890	−0.749448	−	0.662064i	$-0.769681\pi$
$691$	−39.4013	−1.49890	−0.749448	+	0.662064i	$0.769681\pi$
$692$	0	0
$693$	−2.07223	−0.0787173
$694$	0	0
$695$	27.6614	1.04926
$696$	0	0
$697$	−7.74787	−0.293472
$698$	0	0
$699$	40.6353	1.53697
$700$	0	0
$701$	−26.3513	−0.995274	−0.497637	−	0.867385i	$-0.665799\pi$
$701$	−26.3513	−0.995274	−0.497637	+	0.867385i	$0.665799\pi$
$702$	0	0
$703$	−47.8892	−1.80617
$704$	0	0
$705$	−38.0832	−1.43430
$706$	0	0
$707$	−91.0724	−3.42513
$708$	0	0
$709$	5.30926	0.199393	0.0996967	−	0.995018i	$-0.468213\pi$
$709$	5.30926	0.199393	0.0996967	+	0.995018i	$0.468213\pi$
$710$	0	0
$711$	−2.61466	−0.0980574
$712$	0	0
$713$	0.905548	0.0339130
$714$	0	0
$715$	−9.99878	−0.373933
$716$	0	0
$717$	−20.7985	−0.776733
$718$	0	0
$719$	22.8059	0.850518	0.425259	−	0.905072i	$-0.360183\pi$
$719$	22.8059	0.850518	0.425259	+	0.905072i	$0.360183\pi$
$720$	0	0
$721$	−21.9762	−0.818436
$722$	0	0
$723$	37.6922	1.40179
$724$	0	0
$725$	2.38221	0.0884731
$726$	0	0
$727$	36.5135	1.35421	0.677106	−	0.735886i	$-0.263234\pi$
$727$	36.5135	1.35421	0.677106	+	0.735886i	$0.263234\pi$
$728$	0	0
$729$	16.2100	0.600369
$730$	0	0
$731$	−31.3208	−1.15844
$732$	0	0
$733$	−37.1438	−1.37194	−0.685968	−	0.727631i	$-0.740621\pi$
$733$	−37.1438	−1.37194	−0.685968	+	0.727631i	$0.740621\pi$
$734$	0	0
$735$	−109.531	−4.04013
$736$	0	0
$737$	0.0317692	0.00117023
$738$	0	0
$739$	12.1613	0.447359	0.223680	−	0.974663i	$-0.428193\pi$
$739$	12.1613	0.447359	0.223680	+	0.974663i	$0.428193\pi$
$740$	0	0
$741$	−66.1359	−2.42956
$742$	0	0
$743$	−38.7987	−1.42338	−0.711692	−	0.702491i	$-0.752071\pi$
$743$	−38.7987	−1.42338	−0.711692	+	0.702491i	$0.752071\pi$
$744$	0	0
$745$	−15.5741	−0.570590
$746$	0	0
$747$	1.24828	0.0456721
$748$	0	0
$749$	34.1205	1.24673
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	1.67196	0.0609295
$754$	0	0
$755$	−11.2287	−0.408654
$756$	0	0
$757$	26.2313	0.953392	0.476696	−	0.879068i	$-0.341834\pi$
$757$	26.2313	0.953392	0.476696	+	0.879068i	$0.341834\pi$
$758$	0	0
$759$	−0.280166	−0.0101694
$760$	0	0
$761$	12.4367	0.450829	0.225414	−	0.974263i	$-0.427626\pi$
$761$	12.4367	0.450829	0.225414	+	0.974263i	$0.427626\pi$
$762$	0	0
$763$	−0.973620	−0.0352474
$764$	0	0
$765$	−13.8786	−0.501781
$766$	0	0
$767$	27.9361	1.00872
$768$	0	0
$769$	−17.7102	−0.638646	−0.319323	−	0.947646i	$-0.603455\pi$
$769$	−17.7102	−0.638646	−0.319323	+	0.947646i	$0.603455\pi$
$770$	0	0
$771$	−29.6008	−1.06605
$772$	0	0
$773$	−7.63739	−0.274698	−0.137349	−	0.990523i	$-0.543858\pi$
$773$	−7.63739	−0.274698	−0.137349	+	0.990523i	$0.543858\pi$
$774$	0	0
$775$	−16.9523	−0.608945
$776$	0	0
$777$	83.3719	2.99095
$778$	0	0
$779$	8.09476	0.290025
$780$	0	0
$781$	−1.66042	−0.0594145
$782$	0	0
$783$	1.94195	0.0693996
$784$	0	0
$785$	−1.22972	−0.0438907
$786$	0	0
$787$	−13.6617	−0.486988	−0.243494	−	0.969902i	$-0.578294\pi$
$787$	−13.6617	−0.486988	−0.243494	+	0.969902i	$0.578294\pi$
$788$	0	0
$789$	−22.4452	−0.799071
$790$	0	0
$791$	−11.7227	−0.416812
$792$	0	0
$793$	2.37596	0.0843728
$794$	0	0
$795$	−6.63094	−0.235175
$796$	0	0
$797$	−20.3806	−0.721920	−0.360960	−	0.932581i	$-0.617551\pi$
$797$	−20.3806	−0.721920	−0.360960	+	0.932581i	$0.617551\pi$
$798$	0	0
$799$	−32.4140	−1.14673
$800$	0	0
$801$	2.62056	0.0925929
$802$	0	0
$803$	−5.82986	−0.205731
$804$	0	0
$805$	−4.42982	−0.156131
$806$	0	0
$807$	63.8939	2.24917
$808$	0	0
$809$	−46.1849	−1.62378	−0.811888	−	0.583814i	$-0.801560\pi$
$809$	−46.1849	−1.62378	−0.811888	+	0.583814i	$0.801560\pi$
$810$	0	0
$811$	8.80867	0.309314	0.154657	−	0.987968i	$-0.450573\pi$
$811$	8.80867	0.309314	0.154657	+	0.987968i	$0.450573\pi$
$812$	0	0
$813$	−50.6894	−1.77775
$814$	0	0
$815$	−22.8518	−0.800464
$816$	0	0
$817$	32.7231	1.14484
$818$	0	0
$819$	24.6126	0.860034
$820$	0	0
$821$	−38.3203	−1.33739	−0.668695	−	0.743537i	$-0.733147\pi$
$821$	−38.3203	−1.33739	−0.668695	+	0.743537i	$0.733147\pi$
$822$	0	0
$823$	13.1385	0.457978	0.228989	−	0.973429i	$-0.426458\pi$
$823$	13.1385	0.457978	0.228989	+	0.973429i	$0.426458\pi$
$824$	0	0
$825$	5.24485	0.182602
$826$	0	0
$827$	13.6209	0.473645	0.236823	−	0.971553i	$-0.423894\pi$
$827$	13.6209	0.473645	0.236823	+	0.971553i	$0.423894\pi$
$828$	0	0
$829$	−29.4948	−1.02440	−0.512199	−	0.858867i	$-0.671169\pi$
$829$	−29.4948	−1.02440	−0.512199	+	0.858867i	$0.671169\pi$
$830$	0	0
$831$	−39.6921	−1.37690
$832$	0	0
$833$	−93.2263	−3.23010
$834$	0	0
$835$	−56.5211	−1.95599
$836$	0	0
$837$	−13.8193	−0.477666
$838$	0	0
$839$	−33.8354	−1.16813	−0.584064	−	0.811707i	$-0.698538\pi$
$839$	−33.8354	−1.16813	−0.584064	+	0.811707i	$0.698538\pi$
$840$	0	0
$841$	−28.7929	−0.992857
$842$	0	0
$843$	−29.0773	−1.00148
$844$	0	0
$845$	77.1711	2.65477
$846$	0	0
$847$	−53.1749	−1.82711
$848$	0	0
$849$	47.2776	1.62256
$850$	0	0
$851$	2.40955	0.0825983
$852$	0	0
$853$	−33.6531	−1.15226	−0.576130	−	0.817358i	$-0.695438\pi$
$853$	−33.6531	−1.15226	−0.576130	+	0.817358i	$0.695438\pi$
$854$	0	0
$855$	14.4999	0.495887
$856$	0	0
$857$	39.6371	1.35398	0.676989	−	0.735993i	$-0.263284\pi$
$857$	39.6371	1.35398	0.676989	+	0.735993i	$0.263284\pi$
$858$	0	0
$859$	−35.0394	−1.19553	−0.597765	−	0.801672i	$-0.703944\pi$
$859$	−35.0394	−1.19553	−0.597765	+	0.801672i	$0.703944\pi$
$860$	0	0
$861$	−14.0924	−0.480269
$862$	0	0
$863$	15.8424	0.539282	0.269641	−	0.962961i	$-0.413095\pi$
$863$	15.8424	0.539282	0.269641	+	0.962961i	$0.413095\pi$
$864$	0	0
$865$	−18.4971	−0.628922
$866$	0	0
$867$	−22.0521	−0.748928
$868$	0	0
$869$	1.64440	0.0557825
$870$	0	0
$871$	−0.377335	−0.0127855
$872$	0	0
$873$	−7.86201	−0.266089
$874$	0	0
$875$	3.71017	0.125426
$876$	0	0
$877$	15.9438	0.538384	0.269192	−	0.963087i	$-0.413243\pi$
$877$	15.9438	0.538384	0.269192	+	0.963087i	$0.413243\pi$
$878$	0	0
$879$	−7.84148	−0.264487
$880$	0	0
$881$	6.45895	0.217608	0.108804	−	0.994063i	$-0.465298\pi$
$881$	6.45895	0.217608	0.108804	+	0.994063i	$0.465298\pi$
$882$	0	0
$883$	−40.9096	−1.37672	−0.688359	−	0.725370i	$-0.741669\pi$
$883$	−40.9096	−1.37672	−0.688359	+	0.725370i	$0.741669\pi$
$884$	0	0
$885$	−28.6521	−0.963131
$886$	0	0
$887$	−3.65881	−0.122851	−0.0614254	−	0.998112i	$-0.519565\pi$
$887$	−3.65881	−0.122851	−0.0614254	+	0.998112i	$0.519565\pi$
$888$	0	0
$889$	98.2153	3.29404
$890$	0	0
$891$	5.53078	0.185288
$892$	0	0
$893$	33.8653	1.13326
$894$	0	0
$895$	60.9811	2.03837
$896$	0	0
$897$	3.32763	0.111106
$898$	0	0
$899$	−1.47405	−0.0491625
$900$	0	0
$901$	−5.64384	−0.188024
$902$	0	0
$903$	−56.9688	−1.89580
$904$	0	0
$905$	80.5052	2.67608
$906$	0	0
$907$	−5.64899	−0.187572	−0.0937859	−	0.995592i	$-0.529897\pi$
$907$	−5.64899	−0.187572	−0.0937859	+	0.995592i	$0.529897\pi$
$908$	0	0
$909$	−14.9991	−0.497488
$910$	0	0
$911$	3.56590	0.118143	0.0590717	−	0.998254i	$-0.481186\pi$
$911$	3.56590	0.118143	0.0590717	+	0.998254i	$0.481186\pi$
$912$	0	0
$913$	−0.785062	−0.0259818
$914$	0	0
$915$	−2.43685	−0.0805599
$916$	0	0
$917$	−3.74057	−0.123524
$918$	0	0
$919$	−36.2726	−1.19652	−0.598261	−	0.801301i	$-0.704142\pi$
$919$	−36.2726	−1.19652	−0.598261	+	0.801301i	$0.704142\pi$
$920$	0	0
$921$	28.6966	0.945587
$922$	0	0
$923$	19.7214	0.649139
$924$	0	0
$925$	−45.1080	−1.48314
$926$	0	0
$927$	−3.61935	−0.118875
$928$	0	0
$929$	22.1686	0.727328	0.363664	−	0.931530i	$-0.381526\pi$
$929$	22.1686	0.727328	0.363664	+	0.931530i	$0.381526\pi$
$930$	0	0
$931$	97.4002	3.19216
$932$	0	0
$933$	−27.1204	−0.887882
$934$	0	0
$935$	8.72845	0.285451
$936$	0	0
$937$	19.9125	0.650514	0.325257	−	0.945626i	$-0.394549\pi$
$937$	19.9125	0.650514	0.325257	+	0.945626i	$0.394549\pi$
$938$	0	0
$939$	−37.0487	−1.20904
$940$	0	0
$941$	57.4472	1.87272	0.936362	−	0.351036i	$-0.114170\pi$
$941$	57.4472	1.87272	0.936362	+	0.351036i	$0.114170\pi$
$942$	0	0
$943$	−0.407288	−0.0132631
$944$	0	0
$945$	67.6023	2.19910
$946$	0	0
$947$	−18.0258	−0.585759	−0.292880	−	0.956149i	$-0.594614\pi$
$947$	−18.0258	−0.585759	−0.292880	+	0.956149i	$0.594614\pi$
$948$	0	0
$949$	69.2435	2.24774
$950$	0	0
$951$	15.4946	0.502446
$952$	0	0
$953$	−20.3571	−0.659430	−0.329715	−	0.944081i	$-0.606953\pi$
$953$	−20.3571	−0.659430	−0.329715	+	0.944081i	$0.606953\pi$
$954$	0	0
$955$	−29.7709	−0.963364
$956$	0	0
$957$	0.456055	0.0147422
$958$	0	0
$959$	−63.8634	−2.06226
$960$	0	0
$961$	−20.5103	−0.661623
$962$	0	0
$963$	5.61944	0.181084
$964$	0	0
$965$	−6.96004	−0.224052
$966$	0	0
$967$	−9.62718	−0.309589	−0.154795	−	0.987947i	$-0.549472\pi$
$967$	−9.62718	−0.309589	−0.154795	+	0.987947i	$0.549472\pi$
$968$	0	0
$969$	57.7334	1.85467
$970$	0	0
$971$	−60.1871	−1.93150	−0.965749	−	0.259480i	$-0.916449\pi$
$971$	−60.1871	−1.93150	−0.965749	+	0.259480i	$0.916449\pi$
$972$	0	0
$973$	42.8230	1.37284
$974$	0	0
$975$	−62.2950	−1.99504
$976$	0	0
$977$	29.6768	0.949445	0.474722	−	0.880136i	$-0.342549\pi$
$977$	29.6768	0.949445	0.474722	+	0.880136i	$0.342549\pi$
$978$	0	0
$979$	−1.64811	−0.0526739
$980$	0	0
$981$	−0.160349	−0.00511956
$982$	0	0
$983$	59.6053	1.90111	0.950557	−	0.310550i	$-0.100513\pi$
$983$	59.6053	1.90111	0.950557	+	0.310550i	$0.100513\pi$
$984$	0	0
$985$	−41.1312	−1.31055
$986$	0	0
$987$	−58.9572	−1.87663
$988$	0	0
$989$	−1.64647	−0.0523546
$990$	0	0
$991$	−36.2628	−1.15193	−0.575963	−	0.817476i	$-0.695373\pi$
$991$	−36.2628	−1.15193	−0.575963	+	0.817476i	$0.695373\pi$
$992$	0	0
$993$	−5.86784	−0.186210
$994$	0	0
$995$	−78.6880	−2.49458
$996$	0	0
$997$	−34.3246	−1.08707	−0.543535	−	0.839386i	$-0.682915\pi$
$997$	−34.3246	−1.08707	−0.543535	+	0.839386i	$0.682915\pi$
$998$	0	0
$999$	−36.7715	−1.16340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.10	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.10	✓	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)

\(f(q)\)	\(=\)	\(q-1.95337 q^{3} +3.19909 q^{5} +4.95256 q^{7} +0.815657 q^{9} +O(q^{10})\)	\(q-1.95337 q^{3} +3.19909 q^{5} +4.95256 q^{7} +0.815657 q^{9} -0.512980 q^{11} +6.09285 q^{13} -6.24901 q^{15} -5.31876 q^{17} +5.55689 q^{19} -9.67418 q^{21} -0.279596 q^{23} +5.23417 q^{25} +4.26683 q^{27} +0.455127 q^{29} -3.23878 q^{31} +1.00204 q^{33} +15.8437 q^{35} -8.61798 q^{37} -11.9016 q^{39} +1.45671 q^{41} +5.88874 q^{43} +2.60936 q^{45} +6.09428 q^{47} +17.5278 q^{49} +10.3895 q^{51} +1.06112 q^{53} -1.64107 q^{55} -10.8547 q^{57} +4.58507 q^{59} +0.389958 q^{61} +4.03959 q^{63} +19.4916 q^{65} -0.0619307 q^{67} +0.546154 q^{69} +3.23682 q^{71} +11.3647 q^{73} -10.2243 q^{75} -2.54056 q^{77} -3.20559 q^{79} -10.7817 q^{81} +1.53040 q^{83} -17.0152 q^{85} -0.889031 q^{87} +3.21282 q^{89} +30.1752 q^{91} +6.32653 q^{93} +17.7770 q^{95} -9.63886 q^{97} -0.418415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 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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