LMFDB - Embedded newform 6008.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.96484 q^{3} +2.30894 q^{5} -3.32968 q^{7} +5.79029 q^{9} +O(q^{10})$	$q-2.96484 q^{3} +2.30894 q^{5} -3.32968 q^{7} +5.79029 q^{9} +2.14673 q^{11} +7.10748 q^{13} -6.84566 q^{15} +1.94537 q^{17} -1.57430 q^{19} +9.87198 q^{21} +1.61890 q^{23} +0.331224 q^{25} -8.27278 q^{27} -0.641396 q^{29} +9.34979 q^{31} -6.36473 q^{33} -7.68805 q^{35} +3.74005 q^{37} -21.0726 q^{39} +6.15103 q^{41} +7.41503 q^{43} +13.3695 q^{45} +8.21995 q^{47} +4.08678 q^{49} -5.76772 q^{51} -3.90788 q^{53} +4.95669 q^{55} +4.66756 q^{57} -4.72476 q^{59} -3.28446 q^{61} -19.2798 q^{63} +16.4108 q^{65} -1.90142 q^{67} -4.79978 q^{69} -7.00941 q^{71} +5.45222 q^{73} -0.982028 q^{75} -7.14794 q^{77} +9.26342 q^{79} +7.15661 q^{81} -4.26867 q^{83} +4.49175 q^{85} +1.90164 q^{87} -5.75983 q^{89} -23.6657 q^{91} -27.7207 q^{93} -3.63498 q^{95} -3.83915 q^{97} +12.4302 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.96484	−1.71175	−0.855876	−	0.517180i	$-0.826982\pi$
$3$	−2.96484	−1.71175	−0.855876	+	0.517180i	$0.826982\pi$
$4$	0	0
$5$	2.30894	1.03259	0.516296	−	0.856410i	$-0.327311\pi$
$5$	2.30894	1.03259	0.516296	+	0.856410i	$0.327311\pi$
$6$	0	0
$7$	−3.32968	−1.25850	−0.629251	−	0.777202i	$-0.716638\pi$
$7$	−3.32968	−1.25850	−0.629251	+	0.777202i	$0.716638\pi$
$8$	0	0
$9$	5.79029	1.93010
$10$	0	0
$11$	2.14673	0.647264	0.323632	−	0.946183i	$-0.395096\pi$
$11$	2.14673	0.647264	0.323632	+	0.946183i	$0.395096\pi$
$12$	0	0
$13$	7.10748	1.97126	0.985631	−	0.168915i	$-0.0540263\pi$
$13$	7.10748	1.97126	0.985631	+	0.168915i	$0.0540263\pi$
$14$	0	0
$15$	−6.84566	−1.76754
$16$	0	0
$17$	1.94537	0.471822	0.235911	−	0.971775i	$-0.424193\pi$
$17$	1.94537	0.471822	0.235911	+	0.971775i	$0.424193\pi$
$18$	0	0
$19$	−1.57430	−0.361170	−0.180585	−	0.983559i	$-0.557799\pi$
$19$	−1.57430	−0.361170	−0.180585	+	0.983559i	$0.557799\pi$
$20$	0	0
$21$	9.87198	2.15424
$22$	0	0
$23$	1.61890	0.337564	0.168782	−	0.985653i	$-0.446017\pi$
$23$	1.61890	0.337564	0.168782	+	0.985653i	$0.446017\pi$
$24$	0	0
$25$	0.331224	0.0662449
$26$	0	0
$27$	−8.27278	−1.59210
$28$	0	0
$29$	−0.641396	−0.119104	−0.0595521	−	0.998225i	$-0.518967\pi$
$29$	−0.641396	−0.119104	−0.0595521	+	0.998225i	$0.518967\pi$
$30$	0	0
$31$	9.34979	1.67927	0.839636	−	0.543149i	$-0.182768\pi$
$31$	9.34979	1.67927	0.839636	+	0.543149i	$0.182768\pi$
$32$	0	0
$33$	−6.36473	−1.10796
$34$	0	0
$35$	−7.68805	−1.29952
$36$	0	0
$37$	3.74005	0.614861	0.307431	−	0.951571i	$-0.400531\pi$
$37$	3.74005	0.614861	0.307431	+	0.951571i	$0.400531\pi$
$38$	0	0
$39$	−21.0726	−3.37431
$40$	0	0
$41$	6.15103	0.960630	0.480315	−	0.877096i	$-0.340522\pi$
$41$	6.15103	0.960630	0.480315	+	0.877096i	$0.340522\pi$
$42$	0	0
$43$	7.41503	1.13078	0.565391	−	0.824823i	$-0.308725\pi$
$43$	7.41503	1.13078	0.565391	+	0.824823i	$0.308725\pi$
$44$	0	0
$45$	13.3695	1.99300
$46$	0	0
$47$	8.21995	1.19900	0.599502	−	0.800374i	$-0.295366\pi$
$47$	8.21995	1.19900	0.599502	+	0.800374i	$0.295366\pi$
$48$	0	0
$49$	4.08678	0.583825
$50$	0	0
$51$	−5.76772	−0.807642
$52$	0	0
$53$	−3.90788	−0.536788	−0.268394	−	0.963309i	$-0.586493\pi$
$53$	−3.90788	−0.536788	−0.268394	+	0.963309i	$0.586493\pi$
$54$	0	0
$55$	4.95669	0.668360
$56$	0	0
$57$	4.66756	0.618234
$58$	0	0
$59$	−4.72476	−0.615112	−0.307556	−	0.951530i	$-0.599511\pi$
$59$	−4.72476	−0.615112	−0.307556	+	0.951530i	$0.599511\pi$
$60$	0	0
$61$	−3.28446	−0.420532	−0.210266	−	0.977644i	$-0.567433\pi$
$61$	−3.28446	−0.420532	−0.210266	+	0.977644i	$0.567433\pi$
$62$	0	0
$63$	−19.2798	−2.42903
$64$	0	0
$65$	16.4108	2.03551
$66$	0	0
$67$	−1.90142	−0.232295	−0.116148	−	0.993232i	$-0.537055\pi$
$67$	−1.90142	−0.232295	−0.116148	+	0.993232i	$0.537055\pi$
$68$	0	0
$69$	−4.79978	−0.577826
$70$	0	0
$71$	−7.00941	−0.831864	−0.415932	−	0.909396i	$-0.636545\pi$
$71$	−7.00941	−0.831864	−0.415932	+	0.909396i	$0.636545\pi$
$72$	0	0
$73$	5.45222	0.638134	0.319067	−	0.947732i	$-0.396631\pi$
$73$	5.45222	0.638134	0.319067	+	0.947732i	$0.396631\pi$
$74$	0	0
$75$	−0.982028	−0.113395
$76$	0	0
$77$	−7.14794	−0.814583
$78$	0	0
$79$	9.26342	1.04222	0.521108	−	0.853491i	$-0.325519\pi$
$79$	9.26342	1.04222	0.521108	+	0.853491i	$0.325519\pi$
$80$	0	0
$81$	7.15661	0.795179
$82$	0	0
$83$	−4.26867	−0.468548	−0.234274	−	0.972171i	$-0.575271\pi$
$83$	−4.26867	−0.468548	−0.234274	+	0.972171i	$0.575271\pi$
$84$	0	0
$85$	4.49175	0.487199
$86$	0	0
$87$	1.90164	0.203877
$88$	0	0
$89$	−5.75983	−0.610541	−0.305270	−	0.952266i	$-0.598747\pi$
$89$	−5.75983	−0.610541	−0.305270	+	0.952266i	$0.598747\pi$
$90$	0	0
$91$	−23.6657	−2.48083
$92$	0	0
$93$	−27.7207	−2.87450
$94$	0	0
$95$	−3.63498	−0.372941
$96$	0	0
$97$	−3.83915	−0.389807	−0.194903	−	0.980822i	$-0.562439\pi$
$97$	−3.83915	−0.389807	−0.194903	+	0.980822i	$0.562439\pi$
$98$	0	0
$99$	12.4302	1.24928
$100$	0	0
$101$	−9.18715	−0.914155	−0.457078	−	0.889427i	$-0.651104\pi$
$101$	−9.18715	−0.914155	−0.457078	+	0.889427i	$0.651104\pi$
$102$	0	0
$103$	−16.4749	−1.62332	−0.811660	−	0.584130i	$-0.801436\pi$
$103$	−16.4749	−1.62332	−0.811660	+	0.584130i	$0.801436\pi$
$104$	0	0
$105$	22.7939	2.22445
$106$	0	0
$107$	3.87884	0.374982	0.187491	−	0.982266i	$-0.439964\pi$
$107$	3.87884	0.374982	0.187491	+	0.982266i	$0.439964\pi$
$108$	0	0
$109$	−7.78957	−0.746106	−0.373053	−	0.927810i	$-0.621689\pi$
$109$	−7.78957	−0.746106	−0.373053	+	0.927810i	$0.621689\pi$
$110$	0	0
$111$	−11.0887	−1.05249
$112$	0	0
$113$	14.4347	1.35791	0.678953	−	0.734181i	$-0.262434\pi$
$113$	14.4347	1.35791	0.678953	+	0.734181i	$0.262434\pi$
$114$	0	0
$115$	3.73795	0.348566
$116$	0	0
$117$	41.1544	3.80473
$118$	0	0
$119$	−6.47747	−0.593788
$120$	0	0
$121$	−6.39154	−0.581049
$122$	0	0
$123$	−18.2368	−1.64436
$124$	0	0
$125$	−10.7799	−0.964187
$126$	0	0
$127$	2.43815	0.216351	0.108175	−	0.994132i	$-0.465499\pi$
$127$	2.43815	0.216351	0.108175	+	0.994132i	$0.465499\pi$
$128$	0	0
$129$	−21.9844	−1.93562
$130$	0	0
$131$	−7.53690	−0.658502	−0.329251	−	0.944242i	$-0.606796\pi$
$131$	−7.53690	−0.658502	−0.329251	+	0.944242i	$0.606796\pi$
$132$	0	0
$133$	5.24193	0.454533
$134$	0	0
$135$	−19.1014	−1.64399
$136$	0	0
$137$	6.28100	0.536622	0.268311	−	0.963332i	$-0.413535\pi$
$137$	6.28100	0.536622	0.268311	+	0.963332i	$0.413535\pi$
$138$	0	0
$139$	6.74449	0.572060	0.286030	−	0.958221i	$-0.407664\pi$
$139$	6.74449	0.572060	0.286030	+	0.958221i	$0.407664\pi$
$140$	0	0
$141$	−24.3709	−2.05240
$142$	0	0
$143$	15.2579	1.27593
$144$	0	0
$145$	−1.48095	−0.122986
$146$	0	0
$147$	−12.1167	−0.999365
$148$	0	0
$149$	11.7319	0.961113	0.480556	−	0.876964i	$-0.340435\pi$
$149$	11.7319	0.961113	0.480556	+	0.876964i	$0.340435\pi$
$150$	0	0
$151$	0.710557	0.0578243	0.0289122	−	0.999582i	$-0.490796\pi$
$151$	0.710557	0.0578243	0.0289122	+	0.999582i	$0.490796\pi$
$152$	0	0
$153$	11.2643	0.910662
$154$	0	0
$155$	21.5882	1.73400
$156$	0	0
$157$	18.0109	1.43743	0.718714	−	0.695306i	$-0.244731\pi$
$157$	18.0109	1.43743	0.718714	+	0.695306i	$0.244731\pi$
$158$	0	0
$159$	11.5862	0.918849
$160$	0	0
$161$	−5.39042	−0.424825
$162$	0	0
$163$	6.53890	0.512166	0.256083	−	0.966655i	$-0.417568\pi$
$163$	6.53890	0.512166	0.256083	+	0.966655i	$0.417568\pi$
$164$	0	0
$165$	−14.6958	−1.14407
$166$	0	0
$167$	−3.76092	−0.291029	−0.145514	−	0.989356i	$-0.546484\pi$
$167$	−3.76092	−0.291029	−0.145514	+	0.989356i	$0.546484\pi$
$168$	0	0
$169$	37.5163	2.88587
$170$	0	0
$171$	−9.11568	−0.697093
$172$	0	0
$173$	−9.71597	−0.738691	−0.369346	−	0.929292i	$-0.620418\pi$
$173$	−9.71597	−0.738691	−0.369346	+	0.929292i	$0.620418\pi$
$174$	0	0
$175$	−1.10287	−0.0833692
$176$	0	0
$177$	14.0082	1.05292
$178$	0	0
$179$	17.6447	1.31883	0.659414	−	0.751780i	$-0.270804\pi$
$179$	17.6447	1.31883	0.659414	+	0.751780i	$0.270804\pi$
$180$	0	0
$181$	−18.2802	−1.35875	−0.679377	−	0.733789i	$-0.737750\pi$
$181$	−18.2802	−1.35875	−0.679377	+	0.733789i	$0.737750\pi$
$182$	0	0
$183$	9.73790	0.719846
$184$	0	0
$185$	8.63558	0.634900
$186$	0	0
$187$	4.17619	0.305393
$188$	0	0
$189$	27.5457	2.00366
$190$	0	0
$191$	16.4119	1.18752	0.593762	−	0.804641i	$-0.297642\pi$
$191$	16.4119	1.18752	0.593762	+	0.804641i	$0.297642\pi$
$192$	0	0
$193$	−22.1718	−1.59596	−0.797981	−	0.602683i	$-0.794099\pi$
$193$	−22.1718	−1.59596	−0.797981	+	0.602683i	$0.794099\pi$
$194$	0	0
$195$	−48.6554	−3.48429
$196$	0	0
$197$	−17.1780	−1.22388	−0.611940	−	0.790904i	$-0.709611\pi$
$197$	−17.1780	−1.22388	−0.611940	+	0.790904i	$0.709611\pi$
$198$	0	0
$199$	−1.79443	−0.127204	−0.0636018	−	0.997975i	$-0.520259\pi$
$199$	−1.79443	−0.127204	−0.0636018	+	0.997975i	$0.520259\pi$
$200$	0	0
$201$	5.63741	0.397632
$202$	0	0
$203$	2.13564	0.149893
$204$	0	0
$205$	14.2024	0.991938
$206$	0	0
$207$	9.37391	0.651531
$208$	0	0
$209$	−3.37961	−0.233772
$210$	0	0
$211$	17.1733	1.18226	0.591131	−	0.806576i	$-0.298682\pi$
$211$	17.1733	1.18226	0.591131	+	0.806576i	$0.298682\pi$
$212$	0	0
$213$	20.7818	1.42394
$214$	0	0
$215$	17.1209	1.16764
$216$	0	0
$217$	−31.1318	−2.11337
$218$	0	0
$219$	−16.1650	−1.09233
$220$	0	0
$221$	13.8267	0.930084
$222$	0	0
$223$	12.4274	0.832200	0.416100	−	0.909319i	$-0.363397\pi$
$223$	12.4274	0.832200	0.416100	+	0.909319i	$0.363397\pi$
$224$	0	0
$225$	1.91789	0.127859
$226$	0	0
$227$	10.5705	0.701589	0.350794	−	0.936453i	$-0.385912\pi$
$227$	10.5705	0.701589	0.350794	+	0.936453i	$0.385912\pi$
$228$	0	0
$229$	−0.689808	−0.0455838	−0.0227919	−	0.999740i	$-0.507256\pi$
$229$	−0.689808	−0.0455838	−0.0227919	+	0.999740i	$0.507256\pi$
$230$	0	0
$231$	21.1925	1.39436
$232$	0	0
$233$	−3.82540	−0.250610	−0.125305	−	0.992118i	$-0.539991\pi$
$233$	−3.82540	−0.250610	−0.125305	+	0.992118i	$0.539991\pi$
$234$	0	0
$235$	18.9794	1.23808
$236$	0	0
$237$	−27.4646	−1.78402
$238$	0	0
$239$	−11.9284	−0.771586	−0.385793	−	0.922585i	$-0.626072\pi$
$239$	−11.9284	−0.771586	−0.385793	+	0.922585i	$0.626072\pi$
$240$	0	0
$241$	−25.6594	−1.65287	−0.826435	−	0.563032i	$-0.809635\pi$
$241$	−25.6594	−1.65287	−0.826435	+	0.563032i	$0.809635\pi$
$242$	0	0
$243$	3.60011	0.230947
$244$	0	0
$245$	9.43614	0.602853
$246$	0	0
$247$	−11.1893	−0.711960
$248$	0	0
$249$	12.6559	0.802038
$250$	0	0
$251$	−6.19074	−0.390756	−0.195378	−	0.980728i	$-0.562593\pi$
$251$	−6.19074	−0.390756	−0.195378	+	0.980728i	$0.562593\pi$
$252$	0	0
$253$	3.47535	0.218493
$254$	0	0
$255$	−13.3173	−0.833964
$256$	0	0
$257$	17.7040	1.10434	0.552172	−	0.833730i	$-0.313799\pi$
$257$	17.7040	1.10434	0.552172	+	0.833730i	$0.313799\pi$
$258$	0	0
$259$	−12.4532	−0.773804
$260$	0	0
$261$	−3.71387	−0.229883
$262$	0	0
$263$	−14.1052	−0.869764	−0.434882	−	0.900488i	$-0.643210\pi$
$263$	−14.1052	−0.869764	−0.434882	+	0.900488i	$0.643210\pi$
$264$	0	0
$265$	−9.02307	−0.554283
$266$	0	0
$267$	17.0770	1.04509
$268$	0	0
$269$	13.5630	0.826948	0.413474	−	0.910516i	$-0.364315\pi$
$269$	13.5630	0.826948	0.413474	+	0.910516i	$0.364315\pi$
$270$	0	0
$271$	−13.9666	−0.848412	−0.424206	−	0.905566i	$-0.639447\pi$
$271$	−13.9666	−0.848412	−0.424206	+	0.905566i	$0.639447\pi$
$272$	0	0
$273$	70.1649	4.24658
$274$	0	0
$275$	0.711050	0.0428779
$276$	0	0
$277$	9.91202	0.595556	0.297778	−	0.954635i	$-0.403755\pi$
$277$	9.91202	0.595556	0.297778	+	0.954635i	$0.403755\pi$
$278$	0	0
$279$	54.1380	3.24116
$280$	0	0
$281$	−6.66988	−0.397892	−0.198946	−	0.980010i	$-0.563752\pi$
$281$	−6.66988	−0.397892	−0.198946	+	0.980010i	$0.563752\pi$
$282$	0	0
$283$	−8.00761	−0.476003	−0.238002	−	0.971265i	$-0.576492\pi$
$283$	−8.00761	−0.476003	−0.238002	+	0.971265i	$0.576492\pi$
$284$	0	0
$285$	10.7771	0.638383
$286$	0	0
$287$	−20.4810	−1.20895
$288$	0	0
$289$	−13.2155	−0.777384
$290$	0	0
$291$	11.3825	0.667253
$292$	0	0
$293$	0.724694	0.0423371	0.0211685	−	0.999776i	$-0.493261\pi$
$293$	0.724694	0.0423371	0.0211685	+	0.999776i	$0.493261\pi$
$294$	0	0
$295$	−10.9092	−0.635159
$296$	0	0
$297$	−17.7594	−1.03051
$298$	0	0
$299$	11.5063	0.665427
$300$	0	0
$301$	−24.6897	−1.42309
$302$	0	0
$303$	27.2384	1.56481
$304$	0	0
$305$	−7.58363	−0.434237
$306$	0	0
$307$	14.0225	0.800306	0.400153	−	0.916448i	$-0.368957\pi$
$307$	14.0225	0.800306	0.400153	+	0.916448i	$0.368957\pi$
$308$	0	0
$309$	48.8455	2.77872
$310$	0	0
$311$	−23.4147	−1.32773	−0.663863	−	0.747854i	$-0.731084\pi$
$311$	−23.4147	−1.32773	−0.663863	+	0.747854i	$0.731084\pi$
$312$	0	0
$313$	−32.4687	−1.83524	−0.917621	−	0.397457i	$-0.869893\pi$
$313$	−32.4687	−1.83524	−0.917621	+	0.397457i	$0.869893\pi$
$314$	0	0
$315$	−44.5161	−2.50820
$316$	0	0
$317$	25.6751	1.44206	0.721028	−	0.692905i	$-0.243670\pi$
$317$	25.6751	1.44206	0.721028	+	0.692905i	$0.243670\pi$
$318$	0	0
$319$	−1.37691	−0.0770919
$320$	0	0
$321$	−11.5002	−0.641877
$322$	0	0
$323$	−3.06260	−0.170408
$324$	0	0
$325$	2.35417	0.130586
$326$	0	0
$327$	23.0949	1.27715
$328$	0	0
$329$	−27.3698	−1.50895
$330$	0	0
$331$	−12.9087	−0.709526	−0.354763	−	0.934956i	$-0.615438\pi$
$331$	−12.9087	−0.709526	−0.354763	+	0.934956i	$0.615438\pi$
$332$	0	0
$333$	21.6560	1.18674
$334$	0	0
$335$	−4.39027	−0.239866
$336$	0	0
$337$	32.5660	1.77398	0.886990	−	0.461788i	$-0.152792\pi$
$337$	32.5660	1.77398	0.886990	+	0.461788i	$0.152792\pi$
$338$	0	0
$339$	−42.7968	−2.32440
$340$	0	0
$341$	20.0715	1.08693
$342$	0	0
$343$	9.70010	0.523756
$344$	0	0
$345$	−11.0824	−0.596658
$346$	0	0
$347$	22.8921	1.22891	0.614455	−	0.788952i	$-0.289376\pi$
$347$	22.8921	1.22891	0.614455	+	0.788952i	$0.289376\pi$
$348$	0	0
$349$	22.8101	1.22100	0.610498	−	0.792018i	$-0.290969\pi$
$349$	22.8101	1.22100	0.610498	+	0.792018i	$0.290969\pi$
$350$	0	0
$351$	−58.7986	−3.13844
$352$	0	0
$353$	34.5706	1.84001	0.920003	−	0.391911i	$-0.128186\pi$
$353$	34.5706	1.84001	0.920003	+	0.391911i	$0.128186\pi$
$354$	0	0
$355$	−16.1843	−0.858975
$356$	0	0
$357$	19.2047	1.01642
$358$	0	0
$359$	29.5296	1.55851	0.779256	−	0.626706i	$-0.215597\pi$
$359$	29.5296	1.55851	0.779256	+	0.626706i	$0.215597\pi$
$360$	0	0
$361$	−16.5216	−0.869556
$362$	0	0
$363$	18.9499	0.994612
$364$	0	0
$365$	12.5889	0.658931
$366$	0	0
$367$	36.9416	1.92833	0.964167	−	0.265296i	$-0.0854698\pi$
$367$	36.9416	1.92833	0.964167	+	0.265296i	$0.0854698\pi$
$368$	0	0
$369$	35.6163	1.85411
$370$	0	0
$371$	13.0120	0.675549
$372$	0	0
$373$	5.19099	0.268779	0.134390	−	0.990929i	$-0.457093\pi$
$373$	5.19099	0.268779	0.134390	+	0.990929i	$0.457093\pi$
$374$	0	0
$375$	31.9608	1.65045
$376$	0	0
$377$	−4.55871	−0.234785
$378$	0	0
$379$	18.3266	0.941376	0.470688	−	0.882300i	$-0.344006\pi$
$379$	18.3266	0.941376	0.470688	+	0.882300i	$0.344006\pi$
$380$	0	0
$381$	−7.22872	−0.370339
$382$	0	0
$383$	29.8558	1.52556	0.762779	−	0.646659i	$-0.223834\pi$
$383$	29.8558	1.52556	0.762779	+	0.646659i	$0.223834\pi$
$384$	0	0
$385$	−16.5042	−0.841131
$386$	0	0
$387$	42.9352	2.18252
$388$	0	0
$389$	21.3831	1.08417	0.542084	−	0.840324i	$-0.317635\pi$
$389$	21.3831	1.08417	0.542084	+	0.840324i	$0.317635\pi$
$390$	0	0
$391$	3.14936	0.159270
$392$	0	0
$393$	22.3457	1.12719
$394$	0	0
$395$	21.3887	1.07618
$396$	0	0
$397$	18.1874	0.912799	0.456400	−	0.889775i	$-0.349139\pi$
$397$	18.1874	0.912799	0.456400	+	0.889775i	$0.349139\pi$
$398$	0	0
$399$	−15.5415	−0.778048
$400$	0	0
$401$	−8.76900	−0.437903	−0.218952	−	0.975736i	$-0.570264\pi$
$401$	−8.76900	−0.437903	−0.218952	+	0.975736i	$0.570264\pi$
$402$	0	0
$403$	66.4535	3.31028
$404$	0	0
$405$	16.5242	0.821095
$406$	0	0
$407$	8.02890	0.397978
$408$	0	0
$409$	14.3029	0.707231	0.353616	−	0.935391i	$-0.384952\pi$
$409$	14.3029	0.707231	0.353616	+	0.935391i	$0.384952\pi$
$410$	0	0
$411$	−18.6222	−0.918564
$412$	0	0
$413$	15.7320	0.774119
$414$	0	0
$415$	−9.85613	−0.483818
$416$	0	0
$417$	−19.9963	−0.979225
$418$	0	0
$419$	−16.0911	−0.786103	−0.393052	−	0.919516i	$-0.628581\pi$
$419$	−16.0911	−0.786103	−0.393052	+	0.919516i	$0.628581\pi$
$420$	0	0
$421$	6.01049	0.292934	0.146467	−	0.989216i	$-0.453210\pi$
$421$	6.01049	0.292934	0.146467	+	0.989216i	$0.453210\pi$
$422$	0	0
$423$	47.5959	2.31419
$424$	0	0
$425$	0.644354	0.0312558
$426$	0	0
$427$	10.9362	0.529240
$428$	0	0
$429$	−45.2372	−2.18407
$430$	0	0
$431$	−28.4861	−1.37213	−0.686064	−	0.727541i	$-0.740663\pi$
$431$	−28.4861	−1.37213	−0.686064	+	0.727541i	$0.740663\pi$
$432$	0	0
$433$	−33.5031	−1.61006	−0.805028	−	0.593237i	$-0.797850\pi$
$433$	−33.5031	−1.61006	−0.805028	+	0.593237i	$0.797850\pi$
$434$	0	0
$435$	4.39077	0.210522
$436$	0	0
$437$	−2.54864	−0.121918
$438$	0	0
$439$	−19.3704	−0.924498	−0.462249	−	0.886750i	$-0.652957\pi$
$439$	−19.3704	−0.924498	−0.462249	+	0.886750i	$0.652957\pi$
$440$	0	0
$441$	23.6636	1.12684
$442$	0	0
$443$	−13.6945	−0.650647	−0.325324	−	0.945603i	$-0.605473\pi$
$443$	−13.6945	−0.650647	−0.325324	+	0.945603i	$0.605473\pi$
$444$	0	0
$445$	−13.2991	−0.630439
$446$	0	0
$447$	−34.7832	−1.64519
$448$	0	0
$449$	40.0748	1.89125	0.945623	−	0.325265i	$-0.105454\pi$
$449$	40.0748	1.89125	0.945623	+	0.325265i	$0.105454\pi$
$450$	0	0
$451$	13.2046	0.621782
$452$	0	0
$453$	−2.10669	−0.0989809
$454$	0	0
$455$	−54.6427	−2.56169
$456$	0	0
$457$	5.56196	0.260178	0.130089	−	0.991502i	$-0.458474\pi$
$457$	5.56196	0.260178	0.130089	+	0.991502i	$0.458474\pi$
$458$	0	0
$459$	−16.0936	−0.751186
$460$	0	0
$461$	22.3505	1.04097	0.520484	−	0.853872i	$-0.325752\pi$
$461$	22.3505	1.04097	0.520484	+	0.853872i	$0.325752\pi$
$462$	0	0
$463$	10.9988	0.511157	0.255578	−	0.966788i	$-0.417734\pi$
$463$	10.9988	0.511157	0.255578	+	0.966788i	$0.417734\pi$
$464$	0	0
$465$	−64.0055	−2.96818
$466$	0	0
$467$	13.2253	0.611992	0.305996	−	0.952033i	$-0.401011\pi$
$467$	13.2253	0.611992	0.305996	+	0.952033i	$0.401011\pi$
$468$	0	0
$469$	6.33112	0.292344
$470$	0	0
$471$	−53.3996	−2.46052
$472$	0	0
$473$	15.9181	0.731915
$474$	0	0
$475$	−0.521448	−0.0239257
$476$	0	0
$477$	−22.6277	−1.03605
$478$	0	0
$479$	−17.2571	−0.788496	−0.394248	−	0.919004i	$-0.628995\pi$
$479$	−17.2571	−0.788496	−0.394248	+	0.919004i	$0.628995\pi$
$480$	0	0
$481$	26.5824	1.21205
$482$	0	0
$483$	15.9818	0.727195
$484$	0	0
$485$	−8.86439	−0.402511
$486$	0	0
$487$	−35.0380	−1.58772	−0.793861	−	0.608099i	$-0.791932\pi$
$487$	−35.0380	−1.58772	−0.793861	+	0.608099i	$0.791932\pi$
$488$	0	0
$489$	−19.3868	−0.876702
$490$	0	0
$491$	−17.1423	−0.773621	−0.386810	−	0.922159i	$-0.626423\pi$
$491$	−17.1423	−0.773621	−0.386810	+	0.922159i	$0.626423\pi$
$492$	0	0
$493$	−1.24775	−0.0561959
$494$	0	0
$495$	28.7007	1.29000
$496$	0	0
$497$	23.3391	1.04690
$498$	0	0
$499$	9.99835	0.447588	0.223794	−	0.974637i	$-0.428156\pi$
$499$	9.99835	0.447588	0.223794	+	0.974637i	$0.428156\pi$
$500$	0	0
$501$	11.1505	0.498170
$502$	0	0
$503$	23.8412	1.06303	0.531514	−	0.847049i	$-0.321623\pi$
$503$	23.8412	1.06303	0.531514	+	0.847049i	$0.321623\pi$
$504$	0	0
$505$	−21.2126	−0.943949
$506$	0	0
$507$	−111.230	−4.93990
$508$	0	0
$509$	9.69965	0.429929	0.214965	−	0.976622i	$-0.431036\pi$
$509$	9.69965	0.429929	0.214965	+	0.976622i	$0.431036\pi$
$510$	0	0
$511$	−18.1541	−0.803092
$512$	0	0
$513$	13.0239	0.575018
$514$	0	0
$515$	−38.0396	−1.67623
$516$	0	0
$517$	17.6460	0.776072
$518$	0	0
$519$	28.8063	1.26446
$520$	0	0
$521$	36.3748	1.59361	0.796803	−	0.604239i	$-0.206523\pi$
$521$	36.3748	1.59361	0.796803	+	0.604239i	$0.206523\pi$
$522$	0	0
$523$	−1.54998	−0.0677759	−0.0338879	−	0.999426i	$-0.510789\pi$
$523$	−1.54998	−0.0677759	−0.0338879	+	0.999426i	$0.510789\pi$
$524$	0	0
$525$	3.26984	0.142708
$526$	0	0
$527$	18.1888	0.792317
$528$	0	0
$529$	−20.3792	−0.886051
$530$	0	0
$531$	−27.3578	−1.18723
$532$	0	0
$533$	43.7184	1.89365
$534$	0	0
$535$	8.95604	0.387203
$536$	0	0
$537$	−52.3138	−2.25751
$538$	0	0
$539$	8.77322	0.377889
$540$	0	0
$541$	15.7180	0.675771	0.337885	−	0.941187i	$-0.390288\pi$
$541$	15.7180	0.675771	0.337885	+	0.941187i	$0.390288\pi$
$542$	0	0
$543$	54.1978	2.32585
$544$	0	0
$545$	−17.9857	−0.770422
$546$	0	0
$547$	30.1450	1.28891	0.644454	−	0.764643i	$-0.277085\pi$
$547$	30.1450	1.28891	0.644454	+	0.764643i	$0.277085\pi$
$548$	0	0
$549$	−19.0180	−0.811667
$550$	0	0
$551$	1.00975	0.0430169
$552$	0	0
$553$	−30.8442	−1.31163
$554$	0	0
$555$	−25.6031	−1.08679
$556$	0	0
$557$	−33.5264	−1.42056	−0.710279	−	0.703921i	$-0.751431\pi$
$557$	−33.5264	−1.42056	−0.710279	+	0.703921i	$0.751431\pi$
$558$	0	0
$559$	52.7022	2.22907
$560$	0	0
$561$	−12.3818	−0.522758
$562$	0	0
$563$	−32.3019	−1.36136	−0.680681	−	0.732580i	$-0.738316\pi$
$563$	−32.3019	−1.36136	−0.680681	+	0.732580i	$0.738316\pi$
$564$	0	0
$565$	33.3290	1.40216
$566$	0	0
$567$	−23.8292	−1.00073
$568$	0	0
$569$	−43.7160	−1.83267	−0.916335	−	0.400412i	$-0.868867\pi$
$569$	−43.7160	−1.83267	−0.916335	+	0.400412i	$0.868867\pi$
$570$	0	0
$571$	36.1720	1.51375	0.756876	−	0.653558i	$-0.226725\pi$
$571$	36.1720	1.51375	0.756876	+	0.653558i	$0.226725\pi$
$572$	0	0
$573$	−48.6587	−2.03275
$574$	0	0
$575$	0.536219	0.0223619
$576$	0	0
$577$	17.9014	0.745246	0.372623	−	0.927983i	$-0.378458\pi$
$577$	17.9014	0.745246	0.372623	+	0.927983i	$0.378458\pi$
$578$	0	0
$579$	65.7359	2.73189
$580$	0	0
$581$	14.2133	0.589668
$582$	0	0
$583$	−8.38917	−0.347444
$584$	0	0
$585$	95.0232	3.92873
$586$	0	0
$587$	5.26648	0.217371	0.108686	−	0.994076i	$-0.465336\pi$
$587$	5.26648	0.217371	0.108686	+	0.994076i	$0.465336\pi$
$588$	0	0
$589$	−14.7194	−0.606503
$590$	0	0
$591$	50.9300	2.09498
$592$	0	0
$593$	7.47905	0.307128	0.153564	−	0.988139i	$-0.450925\pi$
$593$	7.47905	0.307128	0.153564	+	0.988139i	$0.450925\pi$
$594$	0	0
$595$	−14.9561	−0.613141
$596$	0	0
$597$	5.32020	0.217741
$598$	0	0
$599$	−10.9421	−0.447081	−0.223541	−	0.974695i	$-0.571762\pi$
$599$	−10.9421	−0.447081	−0.223541	+	0.974695i	$0.571762\pi$
$600$	0	0
$601$	34.1253	1.39200	0.696000	−	0.718042i	$-0.254961\pi$
$601$	34.1253	1.39200	0.696000	+	0.718042i	$0.254961\pi$
$602$	0	0
$603$	−11.0098	−0.448353
$604$	0	0
$605$	−14.7577	−0.599986
$606$	0	0
$607$	8.43853	0.342510	0.171255	−	0.985227i	$-0.445218\pi$
$607$	8.43853	0.342510	0.171255	+	0.985227i	$0.445218\pi$
$608$	0	0
$609$	−6.33185	−0.256579
$610$	0	0
$611$	58.4232	2.36355
$612$	0	0
$613$	−18.9245	−0.764353	−0.382177	−	0.924089i	$-0.624825\pi$
$613$	−18.9245	−0.764353	−0.382177	+	0.924089i	$0.624825\pi$
$614$	0	0
$615$	−42.1079	−1.69795
$616$	0	0
$617$	24.7639	0.996959	0.498479	−	0.866902i	$-0.333892\pi$
$617$	24.7639	0.996959	0.498479	+	0.866902i	$0.333892\pi$
$618$	0	0
$619$	−37.7193	−1.51607	−0.758034	−	0.652215i	$-0.773840\pi$
$619$	−37.7193	−1.51607	−0.758034	+	0.652215i	$0.773840\pi$
$620$	0	0
$621$	−13.3928	−0.537435
$622$	0	0
$623$	19.1784	0.768366
$624$	0	0
$625$	−26.5464	−1.06186
$626$	0	0
$627$	10.0200	0.400161
$628$	0	0
$629$	7.27579	0.290105
$630$	0	0
$631$	−45.2793	−1.80254	−0.901269	−	0.433259i	$-0.857363\pi$
$631$	−45.2793	−1.80254	−0.901269	+	0.433259i	$0.857363\pi$
$632$	0	0
$633$	−50.9163	−2.02374
$634$	0	0
$635$	5.62954	0.223402
$636$	0	0
$637$	29.0467	1.15087
$638$	0	0
$639$	−40.5865	−1.60558
$640$	0	0
$641$	14.6587	0.578984	0.289492	−	0.957180i	$-0.406514\pi$
$641$	14.6587	0.578984	0.289492	+	0.957180i	$0.406514\pi$
$642$	0	0
$643$	9.97165	0.393243	0.196622	−	0.980479i	$-0.437003\pi$
$643$	9.97165	0.393243	0.196622	+	0.980479i	$0.437003\pi$
$644$	0	0
$645$	−50.7608	−1.99870
$646$	0	0
$647$	−19.3158	−0.759383	−0.379692	−	0.925113i	$-0.623970\pi$
$647$	−19.3158	−0.759383	−0.379692	+	0.925113i	$0.623970\pi$
$648$	0	0
$649$	−10.1428	−0.398140
$650$	0	0
$651$	92.3010	3.61756
$652$	0	0
$653$	−31.6977	−1.24043	−0.620213	−	0.784433i	$-0.712954\pi$
$653$	−31.6977	−1.24043	−0.620213	+	0.784433i	$0.712954\pi$
$654$	0	0
$655$	−17.4023	−0.679963
$656$	0	0
$657$	31.5699	1.23166
$658$	0	0
$659$	11.2622	0.438712	0.219356	−	0.975645i	$-0.429604\pi$
$659$	11.2622	0.438712	0.219356	+	0.975645i	$0.429604\pi$
$660$	0	0
$661$	−18.0861	−0.703467	−0.351733	−	0.936100i	$-0.614408\pi$
$661$	−18.0861	−0.703467	−0.351733	+	0.936100i	$0.614408\pi$
$662$	0	0
$663$	−40.9940	−1.59207
$664$	0	0
$665$	12.1033	0.469347
$666$	0	0
$667$	−1.03836	−0.0402053
$668$	0	0
$669$	−36.8453	−1.42452
$670$	0	0
$671$	−7.05085	−0.272195
$672$	0	0
$673$	23.3041	0.898306	0.449153	−	0.893455i	$-0.351726\pi$
$673$	23.3041	0.898306	0.449153	+	0.893455i	$0.351726\pi$
$674$	0	0
$675$	−2.74014	−0.105468
$676$	0	0
$677$	9.89149	0.380161	0.190080	−	0.981769i	$-0.439125\pi$
$677$	9.89149	0.380161	0.190080	+	0.981769i	$0.439125\pi$
$678$	0	0
$679$	12.7832	0.490572
$680$	0	0
$681$	−31.3399	−1.20095
$682$	0	0
$683$	6.47860	0.247897	0.123948	−	0.992289i	$-0.460444\pi$
$683$	6.47860	0.247897	0.123948	+	0.992289i	$0.460444\pi$
$684$	0	0
$685$	14.5025	0.554111
$686$	0	0
$687$	2.04517	0.0780282
$688$	0	0
$689$	−27.7752	−1.05815
$690$	0	0
$691$	22.6049	0.859932	0.429966	−	0.902845i	$-0.358525\pi$
$691$	22.6049	0.859932	0.429966	+	0.902845i	$0.358525\pi$
$692$	0	0
$693$	−41.3886	−1.57222
$694$	0	0
$695$	15.5726	0.590704
$696$	0	0
$697$	11.9660	0.453246
$698$	0	0
$699$	11.3417	0.428983
$700$	0	0
$701$	−5.85443	−0.221119	−0.110559	−	0.993870i	$-0.535264\pi$
$701$	−5.85443	−0.221119	−0.110559	+	0.993870i	$0.535264\pi$
$702$	0	0
$703$	−5.88798	−0.222069
$704$	0	0
$705$	−56.2710	−2.11929
$706$	0	0
$707$	30.5903	1.15047
$708$	0	0
$709$	−39.5665	−1.48595	−0.742975	−	0.669319i	$-0.766586\pi$
$709$	−39.5665	−1.48595	−0.742975	+	0.669319i	$0.766586\pi$
$710$	0	0
$711$	53.6379	2.01158
$712$	0	0
$713$	15.1364	0.566862
$714$	0	0
$715$	35.2296	1.31751
$716$	0	0
$717$	35.3659	1.32077
$718$	0	0
$719$	25.1305	0.937211	0.468606	−	0.883407i	$-0.344757\pi$
$719$	25.1305	0.937211	0.468606	+	0.883407i	$0.344757\pi$
$720$	0	0
$721$	54.8562	2.04295
$722$	0	0
$723$	76.0762	2.82930
$724$	0	0
$725$	−0.212446	−0.00789004
$726$	0	0
$727$	−11.6844	−0.433351	−0.216675	−	0.976244i	$-0.569521\pi$
$727$	−11.6844	−0.433351	−0.216675	+	0.976244i	$0.569521\pi$
$728$	0	0
$729$	−32.1436	−1.19050
$730$	0	0
$731$	14.4250	0.533528
$732$	0	0
$733$	−8.64586	−0.319342	−0.159671	−	0.987170i	$-0.551043\pi$
$733$	−8.64586	−0.319342	−0.159671	+	0.987170i	$0.551043\pi$
$734$	0	0
$735$	−27.9767	−1.03194
$736$	0	0
$737$	−4.08184	−0.150357
$738$	0	0
$739$	−13.1160	−0.482480	−0.241240	−	0.970466i	$-0.577554\pi$
$739$	−13.1160	−0.482480	−0.241240	+	0.970466i	$0.577554\pi$
$740$	0	0
$741$	33.1746	1.21870
$742$	0	0
$743$	−35.5580	−1.30450	−0.652249	−	0.758005i	$-0.726174\pi$
$743$	−35.5580	−1.30450	−0.652249	+	0.758005i	$0.726174\pi$
$744$	0	0
$745$	27.0883	0.992437
$746$	0	0
$747$	−24.7169	−0.904343
$748$	0	0
$749$	−12.9153	−0.471915
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	18.3546	0.668878
$754$	0	0
$755$	1.64064	0.0597089
$756$	0	0
$757$	13.7047	0.498105	0.249053	−	0.968490i	$-0.419881\pi$
$757$	13.7047	0.498105	0.249053	+	0.968490i	$0.419881\pi$
$758$	0	0
$759$	−10.3039	−0.374006
$760$	0	0
$761$	−18.2910	−0.663047	−0.331524	−	0.943447i	$-0.607563\pi$
$761$	−18.2910	−0.663047	−0.331524	+	0.943447i	$0.607563\pi$
$762$	0	0
$763$	25.9368	0.938975
$764$	0	0
$765$	26.0086	0.940342
$766$	0	0
$767$	−33.5812	−1.21255
$768$	0	0
$769$	7.69104	0.277346	0.138673	−	0.990338i	$-0.455716\pi$
$769$	7.69104	0.277346	0.138673	+	0.990338i	$0.455716\pi$
$770$	0	0
$771$	−52.4896	−1.89037
$772$	0	0
$773$	18.4709	0.664353	0.332177	−	0.943217i	$-0.392217\pi$
$773$	18.4709	0.664353	0.332177	+	0.943217i	$0.392217\pi$
$774$	0	0
$775$	3.09688	0.111243
$776$	0	0
$777$	36.9218	1.32456
$778$	0	0
$779$	−9.68360	−0.346951
$780$	0	0
$781$	−15.0473	−0.538436
$782$	0	0
$783$	5.30612	0.189625
$784$	0	0
$785$	41.5862	1.48428
$786$	0	0
$787$	12.2391	0.436276	0.218138	−	0.975918i	$-0.430002\pi$
$787$	12.2391	0.436276	0.218138	+	0.975918i	$0.430002\pi$
$788$	0	0
$789$	41.8197	1.48882
$790$	0	0
$791$	−48.0631	−1.70893
$792$	0	0
$793$	−23.3442	−0.828978
$794$	0	0
$795$	26.7520	0.948795
$796$	0	0
$797$	41.6753	1.47621	0.738107	−	0.674683i	$-0.235720\pi$
$797$	41.6753	1.47621	0.738107	+	0.674683i	$0.235720\pi$
$798$	0	0
$799$	15.9909	0.565716
$800$	0	0
$801$	−33.3511	−1.17840
$802$	0	0
$803$	11.7045	0.413041
$804$	0	0
$805$	−12.4462	−0.438670
$806$	0	0
$807$	−40.2120	−1.41553
$808$	0	0
$809$	−38.9670	−1.37001	−0.685004	−	0.728539i	$-0.740200\pi$
$809$	−38.9670	−1.37001	−0.685004	+	0.728539i	$0.740200\pi$
$810$	0	0
$811$	12.1814	0.427748	0.213874	−	0.976861i	$-0.431392\pi$
$811$	12.1814	0.427748	0.213874	+	0.976861i	$0.431392\pi$
$812$	0	0
$813$	41.4089	1.45227
$814$	0	0
$815$	15.0980	0.528858
$816$	0	0
$817$	−11.6735	−0.408405
$818$	0	0
$819$	−137.031	−4.78825
$820$	0	0
$821$	−1.29476	−0.0451876	−0.0225938	−	0.999745i	$-0.507192\pi$
$821$	−1.29476	−0.0451876	−0.0225938	+	0.999745i	$0.507192\pi$
$822$	0	0
$823$	−7.93827	−0.276711	−0.138355	−	0.990383i	$-0.544182\pi$
$823$	−7.93827	−0.276711	−0.138355	+	0.990383i	$0.544182\pi$
$824$	0	0
$825$	−2.10815	−0.0733964
$826$	0	0
$827$	11.5139	0.400376	0.200188	−	0.979758i	$-0.435845\pi$
$827$	11.5139	0.400376	0.200188	+	0.979758i	$0.435845\pi$
$828$	0	0
$829$	−32.2404	−1.11975	−0.559877	−	0.828576i	$-0.689152\pi$
$829$	−32.2404	−1.11975	−0.559877	+	0.828576i	$0.689152\pi$
$830$	0	0
$831$	−29.3876	−1.01944
$832$	0	0
$833$	7.95030	0.275462
$834$	0	0
$835$	−8.68376	−0.300514
$836$	0	0
$837$	−77.3488	−2.67356
$838$	0	0
$839$	−42.5398	−1.46864	−0.734318	−	0.678805i	$-0.762498\pi$
$839$	−42.5398	−1.46864	−0.734318	+	0.678805i	$0.762498\pi$
$840$	0	0
$841$	−28.5886	−0.985814
$842$	0	0
$843$	19.7752	0.681092
$844$	0	0
$845$	86.6231	2.97993
$846$	0	0
$847$	21.2818	0.731251
$848$	0	0
$849$	23.7413	0.814800
$850$	0	0
$851$	6.05478	0.207555
$852$	0	0
$853$	41.2526	1.41246	0.706231	−	0.707981i	$-0.250394\pi$
$853$	41.2526	1.41246	0.706231	+	0.707981i	$0.250394\pi$
$854$	0	0
$855$	−21.0476	−0.719813
$856$	0	0
$857$	12.3667	0.422437	0.211219	−	0.977439i	$-0.432257\pi$
$857$	12.3667	0.422437	0.211219	+	0.977439i	$0.432257\pi$
$858$	0	0
$859$	0.456325	0.0155696	0.00778480	−	0.999970i	$-0.497522\pi$
$859$	0.456325	0.0155696	0.00778480	+	0.999970i	$0.497522\pi$
$860$	0	0
$861$	60.7229	2.06943
$862$	0	0
$863$	−0.736221	−0.0250613	−0.0125306	−	0.999921i	$-0.503989\pi$
$863$	−0.736221	−0.0250613	−0.0125306	+	0.999921i	$0.503989\pi$
$864$	0	0
$865$	−22.4336	−0.762766
$866$	0	0
$867$	39.1820	1.33069
$868$	0	0
$869$	19.8861	0.674589
$870$	0	0
$871$	−13.5143	−0.457915
$872$	0	0
$873$	−22.2298	−0.752365
$874$	0	0
$875$	35.8938	1.21343
$876$	0	0
$877$	4.86767	0.164369	0.0821847	−	0.996617i	$-0.473810\pi$
$877$	4.86767	0.164369	0.0821847	+	0.996617i	$0.473810\pi$
$878$	0	0
$879$	−2.14860	−0.0724706
$880$	0	0
$881$	−5.61743	−0.189256	−0.0946279	−	0.995513i	$-0.530166\pi$
$881$	−5.61743	−0.189256	−0.0946279	+	0.995513i	$0.530166\pi$
$882$	0	0
$883$	25.3147	0.851906	0.425953	−	0.904745i	$-0.359939\pi$
$883$	25.3147	0.851906	0.425953	+	0.904745i	$0.359939\pi$
$884$	0	0
$885$	32.3441	1.08724
$886$	0	0
$887$	−48.8485	−1.64017	−0.820086	−	0.572240i	$-0.806075\pi$
$887$	−48.8485	−1.64017	−0.820086	+	0.572240i	$0.806075\pi$
$888$	0	0
$889$	−8.11825	−0.272277
$890$	0	0
$891$	15.3633	0.514691
$892$	0	0
$893$	−12.9407	−0.433044
$894$	0	0
$895$	40.7407	1.36181
$896$	0	0
$897$	−34.1144	−1.13905
$898$	0	0
$899$	−5.99692	−0.200008
$900$	0	0
$901$	−7.60227	−0.253268
$902$	0	0
$903$	73.2011	2.43598
$904$	0	0
$905$	−42.2079	−1.40304
$906$	0	0
$907$	−32.2393	−1.07049	−0.535244	−	0.844697i	$-0.679781\pi$
$907$	−32.2393	−1.07049	−0.535244	+	0.844697i	$0.679781\pi$
$908$	0	0
$909$	−53.1963	−1.76441
$910$	0	0
$911$	−11.6411	−0.385685	−0.192843	−	0.981230i	$-0.561771\pi$
$911$	−11.6411	−0.385685	−0.192843	+	0.981230i	$0.561771\pi$
$912$	0	0
$913$	−9.16370	−0.303274
$914$	0	0
$915$	22.4843	0.743307
$916$	0	0
$917$	25.0955	0.828726
$918$	0	0
$919$	−13.7608	−0.453928	−0.226964	−	0.973903i	$-0.572880\pi$
$919$	−13.7608	−0.453928	−0.226964	+	0.973903i	$0.572880\pi$
$920$	0	0
$921$	−41.5745	−1.36993
$922$	0	0
$923$	−49.8193	−1.63982
$924$	0	0
$925$	1.23880	0.0407314
$926$	0	0
$927$	−95.3945	−3.13317
$928$	0	0
$929$	−1.60249	−0.0525759	−0.0262880	−	0.999654i	$-0.508369\pi$
$929$	−1.60249	−0.0525759	−0.0262880	+	0.999654i	$0.508369\pi$
$930$	0	0
$931$	−6.43383	−0.210860
$932$	0	0
$933$	69.4209	2.27274
$934$	0	0
$935$	9.64260	0.315347
$936$	0	0
$937$	48.8298	1.59520	0.797599	−	0.603188i	$-0.206103\pi$
$937$	48.8298	1.59520	0.797599	+	0.603188i	$0.206103\pi$
$938$	0	0
$939$	96.2647	3.14148
$940$	0	0
$941$	−46.4533	−1.51433	−0.757167	−	0.653221i	$-0.773417\pi$
$941$	−46.4533	−1.51433	−0.757167	+	0.653221i	$0.773417\pi$
$942$	0	0
$943$	9.95791	0.324274
$944$	0	0
$945$	63.6015	2.06896
$946$	0	0
$947$	−44.3975	−1.44273	−0.721363	−	0.692557i	$-0.756484\pi$
$947$	−44.3975	−1.44273	−0.721363	+	0.692557i	$0.756484\pi$
$948$	0	0
$949$	38.7515	1.25793
$950$	0	0
$951$	−76.1226	−2.46845
$952$	0	0
$953$	30.7460	0.995960	0.497980	−	0.867188i	$-0.334075\pi$
$953$	30.7460	0.995960	0.497980	+	0.867188i	$0.334075\pi$
$954$	0	0
$955$	37.8942	1.22623
$956$	0	0
$957$	4.08231	0.131962
$958$	0	0
$959$	−20.9137	−0.675339
$960$	0	0
$961$	56.4186	1.81996
$962$	0	0
$963$	22.4596	0.723752
$964$	0	0
$965$	−51.1935	−1.64798
$966$	0	0
$967$	9.03412	0.290518	0.145259	−	0.989394i	$-0.453599\pi$
$967$	9.03412	0.290518	0.145259	+	0.989394i	$0.453599\pi$
$968$	0	0
$969$	9.08014	0.291696
$970$	0	0
$971$	42.3739	1.35984	0.679922	−	0.733284i	$-0.262014\pi$
$971$	42.3739	1.35984	0.679922	+	0.733284i	$0.262014\pi$
$972$	0	0
$973$	−22.4570	−0.719938
$974$	0	0
$975$	−6.97975	−0.223531
$976$	0	0
$977$	−20.2542	−0.647988	−0.323994	−	0.946059i	$-0.605026\pi$
$977$	−20.2542	−0.647988	−0.323994	+	0.946059i	$0.605026\pi$
$978$	0	0
$979$	−12.3648	−0.395181
$980$	0	0
$981$	−45.1039	−1.44006
$982$	0	0
$983$	8.87659	0.283119	0.141560	−	0.989930i	$-0.454788\pi$
$983$	8.87659	0.283119	0.141560	+	0.989930i	$0.454788\pi$
$984$	0	0
$985$	−39.6630	−1.26377
$986$	0	0
$987$	81.1472	2.58294
$988$	0	0
$989$	12.0042	0.381711
$990$	0	0
$991$	−47.2553	−1.50111	−0.750557	−	0.660806i	$-0.770215\pi$
$991$	−47.2553	−1.50111	−0.750557	+	0.660806i	$0.770215\pi$
$992$	0	0
$993$	38.2722	1.21453
$994$	0	0
$995$	−4.14324	−0.131349
$996$	0	0
$997$	8.82304	0.279429	0.139714	−	0.990192i	$-0.455382\pi$
$997$	8.82304	0.279429	0.139714	+	0.990192i	$0.455382\pi$
$998$	0	0
$999$	−30.9406	−0.978919

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.4	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.4	✓	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-2.96484 q^{3} +2.30894 q^{5} -3.32968 q^{7} +5.79029 q^{9} +O(q^{10})\)	\(q-2.96484 q^{3} +2.30894 q^{5} -3.32968 q^{7} +5.79029 q^{9} +2.14673 q^{11} +7.10748 q^{13} -6.84566 q^{15} +1.94537 q^{17} -1.57430 q^{19} +9.87198 q^{21} +1.61890 q^{23} +0.331224 q^{25} -8.27278 q^{27} -0.641396 q^{29} +9.34979 q^{31} -6.36473 q^{33} -7.68805 q^{35} +3.74005 q^{37} -21.0726 q^{39} +6.15103 q^{41} +7.41503 q^{43} +13.3695 q^{45} +8.21995 q^{47} +4.08678 q^{49} -5.76772 q^{51} -3.90788 q^{53} +4.95669 q^{55} +4.66756 q^{57} -4.72476 q^{59} -3.28446 q^{61} -19.2798 q^{63} +16.4108 q^{65} -1.90142 q^{67} -4.79978 q^{69} -7.00941 q^{71} +5.45222 q^{73} -0.982028 q^{75} -7.14794 q^{77} +9.26342 q^{79} +7.15661 q^{81} -4.26867 q^{83} +4.49175 q^{85} +1.90164 q^{87} -5.75983 q^{89} -23.6657 q^{91} -27.7207 q^{93} -3.63498 q^{95} -3.83915 q^{97} +12.4302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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