LMFDB - Embedded newform 6008.2.a.e.1.25

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			1.25
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)

$f(q)$	$=$	$q-0.0243322 q^{3} +3.64574 q^{5} -4.79791 q^{7} -2.99941 q^{9} +O(q^{10})$	$q-0.0243322 q^{3} +3.64574 q^{5} -4.79791 q^{7} -2.99941 q^{9} +4.87158 q^{11} +2.44401 q^{13} -0.0887089 q^{15} -2.69151 q^{17} -1.00162 q^{19} +0.116744 q^{21} -2.85733 q^{23} +8.29142 q^{25} +0.145979 q^{27} +5.94696 q^{29} +3.27534 q^{31} -0.118536 q^{33} -17.4919 q^{35} +6.54988 q^{37} -0.0594682 q^{39} -8.68111 q^{41} +2.95148 q^{43} -10.9351 q^{45} -10.3118 q^{47} +16.0199 q^{49} +0.0654904 q^{51} +7.10743 q^{53} +17.7605 q^{55} +0.0243715 q^{57} +8.34625 q^{59} +8.25579 q^{61} +14.3909 q^{63} +8.91024 q^{65} +3.82406 q^{67} +0.0695252 q^{69} -12.3349 q^{71} -12.9118 q^{73} -0.201748 q^{75} -23.3734 q^{77} -7.35680 q^{79} +8.99467 q^{81} +4.97516 q^{83} -9.81256 q^{85} -0.144703 q^{87} +10.6648 q^{89} -11.7262 q^{91} -0.0796962 q^{93} -3.65163 q^{95} +11.6015 q^{97} -14.6119 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.0243322	−0.0140482	−0.00702410	−	0.999975i	$-0.502236\pi$
$3$	−0.0243322	−0.0140482	−0.00702410	+	0.999975i	$0.502236\pi$
$4$	0	0
$5$	3.64574	1.63042	0.815212	−	0.579162i	$-0.196620\pi$
$5$	3.64574	1.63042	0.815212	+	0.579162i	$0.196620\pi$
$6$	0	0
$7$	−4.79791	−1.81344	−0.906719	−	0.421735i	$-0.861421\pi$
$7$	−4.79791	−1.81344	−0.906719	+	0.421735i	$0.861421\pi$
$8$	0	0
$9$	−2.99941	−0.999803
$10$	0	0
$11$	4.87158	1.46884	0.734419	−	0.678697i	$-0.237455\pi$
$11$	4.87158	1.46884	0.734419	+	0.678697i	$0.237455\pi$
$12$	0	0
$13$	2.44401	0.677847	0.338924	−	0.940814i	$-0.389937\pi$
$13$	2.44401	0.677847	0.338924	+	0.940814i	$0.389937\pi$
$14$	0	0
$15$	−0.0887089	−0.0229045
$16$	0	0
$17$	−2.69151	−0.652788	−0.326394	−	0.945234i	$-0.605834\pi$
$17$	−2.69151	−0.652788	−0.326394	+	0.945234i	$0.605834\pi$
$18$	0	0
$19$	−1.00162	−0.229787	−0.114893	−	0.993378i	$-0.536653\pi$
$19$	−1.00162	−0.229787	−0.114893	+	0.993378i	$0.536653\pi$
$20$	0	0
$21$	0.116744	0.0254756
$22$	0	0
$23$	−2.85733	−0.595795	−0.297897	−	0.954598i	$-0.596285\pi$
$23$	−2.85733	−0.595795	−0.297897	+	0.954598i	$0.596285\pi$
$24$	0	0
$25$	8.29142	1.65828
$26$	0	0
$27$	0.145979	0.0280936
$28$	0	0
$29$	5.94696	1.10432	0.552161	−	0.833737i	$-0.313803\pi$
$29$	5.94696	1.10432	0.552161	+	0.833737i	$0.313803\pi$
$30$	0	0
$31$	3.27534	0.588268	0.294134	−	0.955764i	$-0.404969\pi$
$31$	3.27534	0.588268	0.294134	+	0.955764i	$0.404969\pi$
$32$	0	0
$33$	−0.118536	−0.0206345
$34$	0	0
$35$	−17.4919	−2.95667
$36$	0	0
$37$	6.54988	1.07679	0.538396	−	0.842692i	$-0.319030\pi$
$37$	6.54988	1.07679	0.538396	+	0.842692i	$0.319030\pi$
$38$	0	0
$39$	−0.0594682	−0.00952254
$40$	0	0
$41$	−8.68111	−1.35576	−0.677881	−	0.735172i	$-0.737101\pi$
$41$	−8.68111	−1.35576	−0.677881	+	0.735172i	$0.737101\pi$
$42$	0	0
$43$	2.95148	0.450096	0.225048	−	0.974348i	$-0.427746\pi$
$43$	2.95148	0.450096	0.225048	+	0.974348i	$0.427746\pi$
$44$	0	0
$45$	−10.9351	−1.63010
$46$	0	0
$47$	−10.3118	−1.50413	−0.752064	−	0.659090i	$-0.770941\pi$
$47$	−10.3118	−1.50413	−0.752064	+	0.659090i	$0.770941\pi$
$48$	0	0
$49$	16.0199	2.28856
$50$	0	0
$51$	0.0654904	0.00917050
$52$	0	0
$53$	7.10743	0.976281	0.488140	−	0.872765i	$-0.337675\pi$
$53$	7.10743	0.976281	0.488140	+	0.872765i	$0.337675\pi$
$54$	0	0
$55$	17.7605	2.39483
$56$	0	0
$57$	0.0243715	0.00322809
$58$	0	0
$59$	8.34625	1.08659	0.543295	−	0.839542i	$-0.317177\pi$
$59$	8.34625	1.08659	0.543295	+	0.839542i	$0.317177\pi$
$60$	0	0
$61$	8.25579	1.05705	0.528523	−	0.848919i	$-0.322746\pi$
$61$	8.25579	1.05705	0.528523	+	0.848919i	$0.322746\pi$
$62$	0	0
$63$	14.3909	1.81308
$64$	0	0
$65$	8.91024	1.10518
$66$	0	0
$67$	3.82406	0.467183	0.233592	−	0.972335i	$-0.424952\pi$
$67$	3.82406	0.467183	0.233592	+	0.972335i	$0.424952\pi$
$68$	0	0
$69$	0.0695252	0.00836985
$70$	0	0
$71$	−12.3349	−1.46389	−0.731943	−	0.681366i	$-0.761386\pi$
$71$	−12.3349	−1.46389	−0.731943	+	0.681366i	$0.761386\pi$
$72$	0	0
$73$	−12.9118	−1.51121	−0.755606	−	0.655026i	$-0.772658\pi$
$73$	−12.9118	−1.51121	−0.755606	+	0.655026i	$0.772658\pi$
$74$	0	0
$75$	−0.201748	−0.0232959
$76$	0	0
$77$	−23.3734	−2.66365
$78$	0	0
$79$	−7.35680	−0.827705	−0.413853	−	0.910344i	$-0.635817\pi$
$79$	−7.35680	−0.827705	−0.413853	+	0.910344i	$0.635817\pi$
$80$	0	0
$81$	8.99467	0.999408
$82$	0	0
$83$	4.97516	0.546094	0.273047	−	0.962001i	$-0.411968\pi$
$83$	4.97516	0.546094	0.273047	+	0.962001i	$0.411968\pi$
$84$	0	0
$85$	−9.81256	−1.06432
$86$	0	0
$87$	−0.144703	−0.0155138
$88$	0	0
$89$	10.6648	1.13046	0.565231	−	0.824933i	$-0.308787\pi$
$89$	10.6648	1.13046	0.565231	+	0.824933i	$0.308787\pi$
$90$	0	0
$91$	−11.7262	−1.22923
$92$	0	0
$93$	−0.0796962	−0.00826411
$94$	0	0
$95$	−3.65163	−0.374650
$96$	0	0
$97$	11.6015	1.17795	0.588976	−	0.808150i	$-0.299531\pi$
$97$	11.6015	1.17795	0.588976	+	0.808150i	$0.299531\pi$
$98$	0	0
$99$	−14.6119	−1.46855
$100$	0	0
$101$	−6.52237	−0.649000	−0.324500	−	0.945886i	$-0.605196\pi$
$101$	−6.52237	−0.649000	−0.324500	+	0.945886i	$0.605196\pi$
$102$	0	0
$103$	16.8178	1.65711	0.828555	−	0.559907i	$-0.189163\pi$
$103$	16.8178	1.65711	0.828555	+	0.559907i	$0.189163\pi$
$104$	0	0
$105$	0.425617	0.0415360
$106$	0	0
$107$	9.67514	0.935331	0.467666	−	0.883905i	$-0.345095\pi$
$107$	9.67514	0.935331	0.467666	+	0.883905i	$0.345095\pi$
$108$	0	0
$109$	−7.64926	−0.732667	−0.366333	−	0.930484i	$-0.619387\pi$
$109$	−7.64926	−0.732667	−0.366333	+	0.930484i	$0.619387\pi$
$110$	0	0
$111$	−0.159373	−0.0151270
$112$	0	0
$113$	11.2794	1.06108	0.530538	−	0.847661i	$-0.321990\pi$
$113$	11.2794	1.06108	0.530538	+	0.847661i	$0.321990\pi$
$114$	0	0
$115$	−10.4171	−0.971398
$116$	0	0
$117$	−7.33059	−0.677714
$118$	0	0
$119$	12.9136	1.18379
$120$	0	0
$121$	12.7323	1.15748
$122$	0	0
$123$	0.211231	0.0190460
$124$	0	0
$125$	11.9996	1.07328
$126$	0	0
$127$	−1.91112	−0.169585	−0.0847924	−	0.996399i	$-0.527023\pi$
$127$	−1.91112	−0.169585	−0.0847924	+	0.996399i	$0.527023\pi$
$128$	0	0
$129$	−0.0718160	−0.00632304
$130$	0	0
$131$	−12.9963	−1.13549	−0.567746	−	0.823204i	$-0.692184\pi$
$131$	−12.9963	−1.13549	−0.567746	+	0.823204i	$0.692184\pi$
$132$	0	0
$133$	4.80566	0.416704
$134$	0	0
$135$	0.532201	0.0458045
$136$	0	0
$137$	10.3973	0.888302	0.444151	−	0.895952i	$-0.353505\pi$
$137$	10.3973	0.888302	0.444151	+	0.895952i	$0.353505\pi$
$138$	0	0
$139$	−0.787016	−0.0667538	−0.0333769	−	0.999443i	$-0.510626\pi$
$139$	−0.787016	−0.0667538	−0.0333769	+	0.999443i	$0.510626\pi$
$140$	0	0
$141$	0.250908	0.0211303
$142$	0	0
$143$	11.9062	0.995648
$144$	0	0
$145$	21.6811	1.80051
$146$	0	0
$147$	−0.389800	−0.0321502
$148$	0	0
$149$	7.95896	0.652023	0.326012	−	0.945366i	$-0.394295\pi$
$149$	7.95896	0.652023	0.326012	+	0.945366i	$0.394295\pi$
$150$	0	0
$151$	10.3320	0.840810	0.420405	−	0.907337i	$-0.361888\pi$
$151$	10.3320	0.840810	0.420405	+	0.907337i	$0.361888\pi$
$152$	0	0
$153$	8.07295	0.652659
$154$	0	0
$155$	11.9410	0.959127
$156$	0	0
$157$	9.45680	0.754735	0.377367	−	0.926064i	$-0.376829\pi$
$157$	9.45680	0.754735	0.377367	+	0.926064i	$0.376829\pi$
$158$	0	0
$159$	−0.172939	−0.0137150
$160$	0	0
$161$	13.7092	1.08044
$162$	0	0
$163$	15.5233	1.21588	0.607940	−	0.793983i	$-0.291996\pi$
$163$	15.5233	1.21588	0.607940	+	0.793983i	$0.291996\pi$
$164$	0	0
$165$	−0.432153	−0.0336430
$166$	0	0
$167$	6.92499	0.535872	0.267936	−	0.963437i	$-0.413658\pi$
$167$	6.92499	0.535872	0.267936	+	0.963437i	$0.413658\pi$
$168$	0	0
$169$	−7.02680	−0.540523
$170$	0	0
$171$	3.00426	0.229741
$172$	0	0
$173$	22.9779	1.74698	0.873488	−	0.486846i	$-0.161853\pi$
$173$	22.9779	1.74698	0.873488	+	0.486846i	$0.161853\pi$
$174$	0	0
$175$	−39.7815	−3.00720
$176$	0	0
$177$	−0.203083	−0.0152646
$178$	0	0
$179$	−13.7459	−1.02741	−0.513707	−	0.857966i	$-0.671728\pi$
$179$	−13.7459	−1.02741	−0.513707	+	0.857966i	$0.671728\pi$
$180$	0	0
$181$	1.57194	0.116842	0.0584209	−	0.998292i	$-0.481393\pi$
$181$	1.57194	0.116842	0.0584209	+	0.998292i	$0.481393\pi$
$182$	0	0
$183$	−0.200881	−0.0148496
$184$	0	0
$185$	23.8791	1.75563
$186$	0	0
$187$	−13.1119	−0.958839
$188$	0	0
$189$	−0.700393	−0.0509461
$190$	0	0
$191$	2.68618	0.194365	0.0971826	−	0.995267i	$-0.469017\pi$
$191$	2.68618	0.194365	0.0971826	+	0.995267i	$0.469017\pi$
$192$	0	0
$193$	23.4781	1.68999	0.844996	−	0.534773i	$-0.179603\pi$
$193$	23.4781	1.68999	0.844996	+	0.534773i	$0.179603\pi$
$194$	0	0
$195$	−0.216806	−0.0155258
$196$	0	0
$197$	26.0903	1.85886	0.929429	−	0.369000i	$-0.120300\pi$
$197$	26.0903	1.85886	0.929429	+	0.369000i	$0.120300\pi$
$198$	0	0
$199$	−11.5604	−0.819493	−0.409747	−	0.912199i	$-0.634383\pi$
$199$	−11.5604	−0.819493	−0.409747	+	0.912199i	$0.634383\pi$
$200$	0	0
$201$	−0.0930478	−0.00656309
$202$	0	0
$203$	−28.5330	−2.00262
$204$	0	0
$205$	−31.6491	−2.21047
$206$	0	0
$207$	8.57030	0.595677
$208$	0	0
$209$	−4.87946	−0.337519
$210$	0	0
$211$	−7.55009	−0.519770	−0.259885	−	0.965640i	$-0.583685\pi$
$211$	−7.55009	−0.519770	−0.259885	+	0.965640i	$0.583685\pi$
$212$	0	0
$213$	0.300136	0.0205650
$214$	0	0
$215$	10.7603	0.733848
$216$	0	0
$217$	−15.7148	−1.06679
$218$	0	0
$219$	0.314173	0.0212298
$220$	0	0
$221$	−6.57810	−0.442491
$222$	0	0
$223$	9.37636	0.627888	0.313944	−	0.949442i	$-0.398350\pi$
$223$	9.37636	0.627888	0.313944	+	0.949442i	$0.398350\pi$
$224$	0	0
$225$	−24.8693	−1.65796
$226$	0	0
$227$	−24.6052	−1.63311	−0.816553	−	0.577270i	$-0.804118\pi$
$227$	−24.6052	−1.63311	−0.816553	+	0.577270i	$0.804118\pi$
$228$	0	0
$229$	−11.8446	−0.782710	−0.391355	−	0.920240i	$-0.627994\pi$
$229$	−11.8446	−0.782710	−0.391355	+	0.920240i	$0.627994\pi$
$230$	0	0
$231$	0.568726	0.0374195
$232$	0	0
$233$	−9.17382	−0.600997	−0.300498	−	0.953782i	$-0.597153\pi$
$233$	−9.17382	−0.600997	−0.300498	+	0.953782i	$0.597153\pi$
$234$	0	0
$235$	−37.5941	−2.45237
$236$	0	0
$237$	0.179007	0.0116278
$238$	0	0
$239$	8.44997	0.546583	0.273291	−	0.961931i	$-0.411888\pi$
$239$	8.44997	0.546583	0.273291	+	0.961931i	$0.411888\pi$
$240$	0	0
$241$	−2.08752	−0.134469	−0.0672344	−	0.997737i	$-0.521418\pi$
$241$	−2.08752	−0.134469	−0.0672344	+	0.997737i	$0.521418\pi$
$242$	0	0
$243$	−0.656797	−0.0421335
$244$	0	0
$245$	58.4045	3.73132
$246$	0	0
$247$	−2.44797	−0.155760
$248$	0	0
$249$	−0.121057	−0.00767165
$250$	0	0
$251$	8.88243	0.560654	0.280327	−	0.959905i	$-0.409557\pi$
$251$	8.88243	0.560654	0.280327	+	0.959905i	$0.409557\pi$
$252$	0	0
$253$	−13.9197	−0.875126
$254$	0	0
$255$	0.238761	0.0149518
$256$	0	0
$257$	16.4306	1.02491	0.512455	−	0.858714i	$-0.328736\pi$
$257$	16.4306	1.02491	0.512455	+	0.858714i	$0.328736\pi$
$258$	0	0
$259$	−31.4257	−1.95270
$260$	0	0
$261$	−17.8374	−1.10410
$262$	0	0
$263$	−5.35827	−0.330405	−0.165202	−	0.986260i	$-0.552828\pi$
$263$	−5.35827	−0.330405	−0.165202	+	0.986260i	$0.552828\pi$
$264$	0	0
$265$	25.9118	1.59175
$266$	0	0
$267$	−0.259497	−0.0158810
$268$	0	0
$269$	4.59388	0.280094	0.140047	−	0.990145i	$-0.455275\pi$
$269$	4.59388	0.280094	0.140047	+	0.990145i	$0.455275\pi$
$270$	0	0
$271$	2.67681	0.162604	0.0813022	−	0.996689i	$-0.474092\pi$
$271$	2.67681	0.162604	0.0813022	+	0.996689i	$0.474092\pi$
$272$	0	0
$273$	0.285323	0.0172685
$274$	0	0
$275$	40.3923	2.43575
$276$	0	0
$277$	18.2938	1.09917	0.549583	−	0.835439i	$-0.314787\pi$
$277$	18.2938	1.09917	0.549583	+	0.835439i	$0.314787\pi$
$278$	0	0
$279$	−9.82407	−0.588152
$280$	0	0
$281$	−18.9625	−1.13121	−0.565605	−	0.824676i	$-0.691357\pi$
$281$	−18.9625	−1.13121	−0.565605	+	0.824676i	$0.691357\pi$
$282$	0	0
$283$	22.6467	1.34620	0.673102	−	0.739549i	$-0.264961\pi$
$283$	22.6467	1.34620	0.673102	+	0.739549i	$0.264961\pi$
$284$	0	0
$285$	0.0888523	0.00526316
$286$	0	0
$287$	41.6512	2.45859
$288$	0	0
$289$	−9.75576	−0.573868
$290$	0	0
$291$	−0.282290	−0.0165481
$292$	0	0
$293$	16.6347	0.971811	0.485905	−	0.874011i	$-0.338490\pi$
$293$	16.6347	0.971811	0.485905	+	0.874011i	$0.338490\pi$
$294$	0	0
$295$	30.4283	1.77160
$296$	0	0
$297$	0.711148	0.0412650
$298$	0	0
$299$	−6.98336	−0.403858
$300$	0	0
$301$	−14.1609	−0.816222
$302$	0	0
$303$	0.158704	0.00911729
$304$	0	0
$305$	30.0984	1.72343
$306$	0	0
$307$	−10.6372	−0.607096	−0.303548	−	0.952816i	$-0.598171\pi$
$307$	−10.6372	−0.607096	−0.303548	+	0.952816i	$0.598171\pi$
$308$	0	0
$309$	−0.409215	−0.0232794
$310$	0	0
$311$	−11.8635	−0.672718	−0.336359	−	0.941734i	$-0.609196\pi$
$311$	−11.8635	−0.672718	−0.336359	+	0.941734i	$0.609196\pi$
$312$	0	0
$313$	−2.23905	−0.126558	−0.0632792	−	0.997996i	$-0.520156\pi$
$313$	−2.23905	−0.126558	−0.0632792	+	0.997996i	$0.520156\pi$
$314$	0	0
$315$	52.4654	2.95609
$316$	0	0
$317$	−6.12302	−0.343903	−0.171952	−	0.985105i	$-0.555007\pi$
$317$	−6.12302	−0.343903	−0.171952	+	0.985105i	$0.555007\pi$
$318$	0	0
$319$	28.9711	1.62207
$320$	0	0
$321$	−0.235417	−0.0131397
$322$	0	0
$323$	2.69586	0.150002
$324$	0	0
$325$	20.2643	1.12406
$326$	0	0
$327$	0.186123	0.0102927
$328$	0	0
$329$	49.4750	2.72764
$330$	0	0
$331$	1.33682	0.0734782	0.0367391	−	0.999325i	$-0.488303\pi$
$331$	1.33682	0.0734782	0.0367391	+	0.999325i	$0.488303\pi$
$332$	0	0
$333$	−19.6457	−1.07658
$334$	0	0
$335$	13.9415	0.761707
$336$	0	0
$337$	−6.04090	−0.329069	−0.164534	−	0.986371i	$-0.552612\pi$
$337$	−6.04090	−0.329069	−0.164534	+	0.986371i	$0.552612\pi$
$338$	0	0
$339$	−0.274452	−0.0149062
$340$	0	0
$341$	15.9561	0.864070
$342$	0	0
$343$	−43.2767	−2.33672
$344$	0	0
$345$	0.253471	0.0136464
$346$	0	0
$347$	12.4870	0.670339	0.335170	−	0.942158i	$-0.391206\pi$
$347$	12.4870	0.670339	0.335170	+	0.942158i	$0.391206\pi$
$348$	0	0
$349$	5.50783	0.294827	0.147414	−	0.989075i	$-0.452905\pi$
$349$	5.50783	0.294827	0.147414	+	0.989075i	$0.452905\pi$
$350$	0	0
$351$	0.356774	0.0190432
$352$	0	0
$353$	14.5045	0.771996	0.385998	−	0.922500i	$-0.373857\pi$
$353$	14.5045	0.771996	0.385998	+	0.922500i	$0.373857\pi$
$354$	0	0
$355$	−44.9699	−2.38675
$356$	0	0
$357$	−0.314217	−0.0166301
$358$	0	0
$359$	−13.1360	−0.693291	−0.346645	−	0.937996i	$-0.612679\pi$
$359$	−13.1360	−0.693291	−0.346645	+	0.937996i	$0.612679\pi$
$360$	0	0
$361$	−17.9968	−0.947198
$362$	0	0
$363$	−0.309805	−0.0162606
$364$	0	0
$365$	−47.0731	−2.46392
$366$	0	0
$367$	3.03788	0.158576	0.0792881	−	0.996852i	$-0.474735\pi$
$367$	3.03788	0.158576	0.0792881	+	0.996852i	$0.474735\pi$
$368$	0	0
$369$	26.0382	1.35549
$370$	0	0
$371$	−34.1008	−1.77043
$372$	0	0
$373$	−36.8424	−1.90763	−0.953813	−	0.300400i	$-0.902880\pi$
$373$	−36.8424	−1.90763	−0.953813	+	0.300400i	$0.902880\pi$
$374$	0	0
$375$	−0.291978	−0.0150777
$376$	0	0
$377$	14.5345	0.748562
$378$	0	0
$379$	−2.91929	−0.149954	−0.0749769	−	0.997185i	$-0.523888\pi$
$379$	−2.91929	−0.149954	−0.0749769	+	0.997185i	$0.523888\pi$
$380$	0	0
$381$	0.0465018	0.00238236
$382$	0	0
$383$	27.9102	1.42615	0.713073	−	0.701089i	$-0.247303\pi$
$383$	27.9102	1.42615	0.713073	+	0.701089i	$0.247303\pi$
$384$	0	0
$385$	−85.2134	−4.34287
$386$	0	0
$387$	−8.85269	−0.450007
$388$	0	0
$389$	21.1181	1.07073	0.535366	−	0.844620i	$-0.320174\pi$
$389$	21.1181	1.07073	0.535366	+	0.844620i	$0.320174\pi$
$390$	0	0
$391$	7.69055	0.388928
$392$	0	0
$393$	0.316229	0.0159516
$394$	0	0
$395$	−26.8210	−1.34951
$396$	0	0
$397$	−27.1386	−1.36205	−0.681023	−	0.732262i	$-0.738465\pi$
$397$	−27.1386	−1.36205	−0.681023	+	0.732262i	$0.738465\pi$
$398$	0	0
$399$	−0.116932	−0.00585394
$400$	0	0
$401$	−11.5854	−0.578549	−0.289275	−	0.957246i	$-0.593414\pi$
$401$	−11.5854	−0.578549	−0.289275	+	0.957246i	$0.593414\pi$
$402$	0	0
$403$	8.00497	0.398756
$404$	0	0
$405$	32.7922	1.62946
$406$	0	0
$407$	31.9083	1.58163
$408$	0	0
$409$	8.58616	0.424558	0.212279	−	0.977209i	$-0.431911\pi$
$409$	8.58616	0.424558	0.212279	+	0.977209i	$0.431911\pi$
$410$	0	0
$411$	−0.252989	−0.0124790
$412$	0	0
$413$	−40.0446	−1.97046
$414$	0	0
$415$	18.1381	0.890366
$416$	0	0
$417$	0.0191498	0.000937771 0
$418$	0	0
$419$	34.5033	1.68559	0.842797	−	0.538231i	$-0.180907\pi$
$419$	34.5033	1.68559	0.842797	+	0.538231i	$0.180907\pi$
$420$	0	0
$421$	−40.1227	−1.95546	−0.977729	−	0.209870i	$-0.932696\pi$
$421$	−40.1227	−1.95546	−0.977729	+	0.209870i	$0.932696\pi$
$422$	0	0
$423$	30.9292	1.50383
$424$	0	0
$425$	−22.3165	−1.08251
$426$	0	0
$427$	−39.6105	−1.91689
$428$	0	0
$429$	−0.289704	−0.0139871
$430$	0	0
$431$	−36.7139	−1.76845	−0.884223	−	0.467064i	$-0.845312\pi$
$431$	−36.7139	−1.76845	−0.884223	+	0.467064i	$0.845312\pi$
$432$	0	0
$433$	−7.07924	−0.340206	−0.170103	−	0.985426i	$-0.554410\pi$
$433$	−7.07924	−0.340206	−0.170103	+	0.985426i	$0.554410\pi$
$434$	0	0
$435$	−0.527548	−0.0252940
$436$	0	0
$437$	2.86195	0.136906
$438$	0	0
$439$	25.0165	1.19397	0.596987	−	0.802251i	$-0.296365\pi$
$439$	25.0165	1.19397	0.596987	+	0.802251i	$0.296365\pi$
$440$	0	0
$441$	−48.0503	−2.28811
$442$	0	0
$443$	−14.0409	−0.667103	−0.333551	−	0.942732i	$-0.608247\pi$
$443$	−14.0409	−0.667103	−0.333551	+	0.942732i	$0.608247\pi$
$444$	0	0
$445$	38.8809	1.84313
$446$	0	0
$447$	−0.193659	−0.00915975
$448$	0	0
$449$	−38.7301	−1.82779	−0.913894	−	0.405953i	$-0.866940\pi$
$449$	−38.7301	−1.82779	−0.913894	+	0.405953i	$0.866940\pi$
$450$	0	0
$451$	−42.2907	−1.99139
$452$	0	0
$453$	−0.251401	−0.0118119
$454$	0	0
$455$	−42.7505	−2.00417
$456$	0	0
$457$	30.0710	1.40666	0.703330	−	0.710864i	$-0.251696\pi$
$457$	30.0710	1.40666	0.703330	+	0.710864i	$0.251696\pi$
$458$	0	0
$459$	−0.392904	−0.0183392
$460$	0	0
$461$	17.9011	0.833738	0.416869	−	0.908967i	$-0.363127\pi$
$461$	17.9011	0.833738	0.416869	+	0.908967i	$0.363127\pi$
$462$	0	0
$463$	31.7350	1.47485	0.737426	−	0.675428i	$-0.236041\pi$
$463$	31.7350	1.47485	0.737426	+	0.675428i	$0.236041\pi$
$464$	0	0
$465$	−0.290552	−0.0134740
$466$	0	0
$467$	−14.6423	−0.677565	−0.338782	−	0.940865i	$-0.610015\pi$
$467$	−14.6423	−0.677565	−0.338782	+	0.940865i	$0.610015\pi$
$468$	0	0
$469$	−18.3475	−0.847209
$470$	0	0
$471$	−0.230105	−0.0106027
$472$	0	0
$473$	14.3784	0.661118
$474$	0	0
$475$	−8.30482	−0.381051
$476$	0	0
$477$	−21.3181	−0.976088
$478$	0	0
$479$	−5.65723	−0.258486	−0.129243	−	0.991613i	$-0.541255\pi$
$479$	−5.65723	−0.258486	−0.129243	+	0.991613i	$0.541255\pi$
$480$	0	0
$481$	16.0080	0.729901
$482$	0	0
$483$	−0.333575	−0.0151782
$484$	0	0
$485$	42.2960	1.92056
$486$	0	0
$487$	−0.782270	−0.0354481	−0.0177240	−	0.999843i	$-0.505642\pi$
$487$	−0.782270	−0.0354481	−0.0177240	+	0.999843i	$0.505642\pi$
$488$	0	0
$489$	−0.377716	−0.0170809
$490$	0	0
$491$	−27.3481	−1.23420	−0.617102	−	0.786884i	$-0.711693\pi$
$491$	−27.3481	−1.23420	−0.617102	+	0.786884i	$0.711693\pi$
$492$	0	0
$493$	−16.0063	−0.720889
$494$	0	0
$495$	−53.2711	−2.39436
$496$	0	0
$497$	59.1818	2.65467
$498$	0	0
$499$	34.6030	1.54904	0.774521	−	0.632548i	$-0.217991\pi$
$499$	34.6030	1.54904	0.774521	+	0.632548i	$0.217991\pi$
$500$	0	0
$501$	−0.168500	−0.00752804
$502$	0	0
$503$	5.14943	0.229602	0.114801	−	0.993389i	$-0.463377\pi$
$503$	5.14943	0.229602	0.114801	+	0.993389i	$0.463377\pi$
$504$	0	0
$505$	−23.7789	−1.05815
$506$	0	0
$507$	0.170977	0.00759337
$508$	0	0
$509$	−15.1072	−0.669616	−0.334808	−	0.942286i	$-0.608671\pi$
$509$	−15.1072	−0.669616	−0.334808	+	0.942286i	$0.608671\pi$
$510$	0	0
$511$	61.9496	2.74049
$512$	0	0
$513$	−0.146215	−0.00645554
$514$	0	0
$515$	61.3134	2.70179
$516$	0	0
$517$	−50.2347	−2.20932
$518$	0	0
$519$	−0.559102	−0.0245419
$520$	0	0
$521$	37.7036	1.65182	0.825912	−	0.563800i	$-0.190661\pi$
$521$	37.7036	1.65182	0.825912	+	0.563800i	$0.190661\pi$
$522$	0	0
$523$	−29.2444	−1.27877	−0.639384	−	0.768887i	$-0.720811\pi$
$523$	−29.2444	−1.27877	−0.639384	+	0.768887i	$0.720811\pi$
$524$	0	0
$525$	0.967970	0.0422457
$526$	0	0
$527$	−8.81561	−0.384014
$528$	0	0
$529$	−14.8357	−0.645029
$530$	0	0
$531$	−25.0338	−1.08638
$532$	0	0
$533$	−21.2168	−0.919000
$534$	0	0
$535$	35.2730	1.52499
$536$	0	0
$537$	0.334467	0.0144333
$538$	0	0
$539$	78.0424	3.36152
$540$	0	0
$541$	−17.8314	−0.766632	−0.383316	−	0.923617i	$-0.625218\pi$
$541$	−17.8314	−0.766632	−0.383316	+	0.923617i	$0.625218\pi$
$542$	0	0
$543$	−0.0382489	−0.00164142
$544$	0	0
$545$	−27.8872	−1.19456
$546$	0	0
$547$	31.1069	1.33004	0.665018	−	0.746828i	$-0.268424\pi$
$547$	31.1069	1.33004	0.665018	+	0.746828i	$0.268424\pi$
$548$	0	0
$549$	−24.7625	−1.05684
$550$	0	0
$551$	−5.95658	−0.253759
$552$	0	0
$553$	35.2973	1.50099
$554$	0	0
$555$	−0.581032	−0.0246634
$556$	0	0
$557$	19.7653	0.837485	0.418742	−	0.908105i	$-0.362471\pi$
$557$	19.7653	0.837485	0.418742	+	0.908105i	$0.362471\pi$
$558$	0	0
$559$	7.21345	0.305097
$560$	0	0
$561$	0.319042	0.0134700
$562$	0	0
$563$	27.7465	1.16938	0.584689	−	0.811258i	$-0.301217\pi$
$563$	27.7465	1.16938	0.584689	+	0.811258i	$0.301217\pi$
$564$	0	0
$565$	41.1217	1.73000
$566$	0	0
$567$	−43.1556	−1.81237
$568$	0	0
$569$	−6.78314	−0.284364	−0.142182	−	0.989841i	$-0.545412\pi$
$569$	−6.78314	−0.284364	−0.142182	+	0.989841i	$0.545412\pi$
$570$	0	0
$571$	24.0983	1.00848	0.504241	−	0.863563i	$-0.331773\pi$
$571$	24.0983	1.00848	0.504241	+	0.863563i	$0.331773\pi$
$572$	0	0
$573$	−0.0653607	−0.00273048
$574$	0	0
$575$	−23.6913	−0.987997
$576$	0	0
$577$	42.3797	1.76429	0.882145	−	0.470979i	$-0.156099\pi$
$577$	42.3797	1.76429	0.882145	+	0.470979i	$0.156099\pi$
$578$	0	0
$579$	−0.571274	−0.0237413
$580$	0	0
$581$	−23.8703	−0.990309
$582$	0	0
$583$	34.6244	1.43400
$584$	0	0
$585$	−26.7254	−1.10496
$586$	0	0
$587$	−12.6438	−0.521867	−0.260933	−	0.965357i	$-0.584030\pi$
$587$	−12.6438	−0.521867	−0.260933	+	0.965357i	$0.584030\pi$
$588$	0	0
$589$	−3.28063	−0.135176
$590$	0	0
$591$	−0.634835	−0.0261136
$592$	0	0
$593$	−9.53221	−0.391441	−0.195721	−	0.980660i	$-0.562705\pi$
$593$	−9.53221	−0.391441	−0.195721	+	0.980660i	$0.562705\pi$
$594$	0	0
$595$	47.0797	1.93008
$596$	0	0
$597$	0.281289	0.0115124
$598$	0	0
$599$	−19.6173	−0.801541	−0.400770	−	0.916179i	$-0.631258\pi$
$599$	−19.6173	−0.801541	−0.400770	+	0.916179i	$0.631258\pi$
$600$	0	0
$601$	−31.4945	−1.28469	−0.642344	−	0.766416i	$-0.722038\pi$
$601$	−31.4945	−1.28469	−0.642344	+	0.766416i	$0.722038\pi$
$602$	0	0
$603$	−11.4699	−0.467091
$604$	0	0
$605$	46.4187	1.88719
$606$	0	0
$607$	−46.8119	−1.90004	−0.950019	−	0.312192i	$-0.898937\pi$
$607$	−46.8119	−1.90004	−0.950019	+	0.312192i	$0.898937\pi$
$608$	0	0
$609$	0.694270	0.0281332
$610$	0	0
$611$	−25.2021	−1.01957
$612$	0	0
$613$	−16.1156	−0.650903	−0.325451	−	0.945559i	$-0.605516\pi$
$613$	−16.1156	−0.650903	−0.325451	+	0.945559i	$0.605516\pi$
$614$	0	0
$615$	0.770092	0.0310531
$616$	0	0
$617$	−11.2073	−0.451190	−0.225595	−	0.974221i	$-0.572433\pi$
$617$	−11.2073	−0.451190	−0.225595	+	0.974221i	$0.572433\pi$
$618$	0	0
$619$	44.2890	1.78012	0.890062	−	0.455839i	$-0.150661\pi$
$619$	44.2890	1.78012	0.890062	+	0.455839i	$0.150661\pi$
$620$	0	0
$621$	−0.417110	−0.0167380
$622$	0	0
$623$	−51.1685	−2.05002
$624$	0	0
$625$	2.29051	0.0916205
$626$	0	0
$627$	0.118728	0.00474154
$628$	0	0
$629$	−17.6291	−0.702917
$630$	0	0
$631$	10.3507	0.412055	0.206027	−	0.978546i	$-0.433946\pi$
$631$	10.3507	0.412055	0.206027	+	0.978546i	$0.433946\pi$
$632$	0	0
$633$	0.183710	0.00730183
$634$	0	0
$635$	−6.96746	−0.276495
$636$	0	0
$637$	39.1529	1.55129
$638$	0	0
$639$	36.9975	1.46360
$640$	0	0
$641$	−17.9641	−0.709538	−0.354769	−	0.934954i	$-0.615441\pi$
$641$	−17.9641	−0.709538	−0.354769	+	0.934954i	$0.615441\pi$
$642$	0	0
$643$	21.8747	0.862655	0.431328	−	0.902195i	$-0.358045\pi$
$643$	21.8747	0.862655	0.431328	+	0.902195i	$0.358045\pi$
$644$	0	0
$645$	−0.261822	−0.0103092
$646$	0	0
$647$	−26.2293	−1.03118	−0.515591	−	0.856835i	$-0.672427\pi$
$647$	−26.2293	−1.03118	−0.515591	+	0.856835i	$0.672427\pi$
$648$	0	0
$649$	40.6595	1.59602
$650$	0	0
$651$	0.382375	0.0149865
$652$	0	0
$653$	−2.85863	−0.111867	−0.0559333	−	0.998435i	$-0.517813\pi$
$653$	−2.85863	−0.111867	−0.0559333	+	0.998435i	$0.517813\pi$
$654$	0	0
$655$	−47.3811	−1.85133
$656$	0	0
$657$	38.7278	1.51091
$658$	0	0
$659$	26.1073	1.01700	0.508498	−	0.861063i	$-0.330201\pi$
$659$	26.1073	1.01700	0.508498	+	0.861063i	$0.330201\pi$
$660$	0	0
$661$	30.3919	1.18211	0.591055	−	0.806631i	$-0.298712\pi$
$661$	30.3919	1.18211	0.591055	+	0.806631i	$0.298712\pi$
$662$	0	0
$663$	0.160060	0.00621620
$664$	0	0
$665$	17.5202	0.679404
$666$	0	0
$667$	−16.9924	−0.657950
$668$	0	0
$669$	−0.228148	−0.00882069
$670$	0	0
$671$	40.2187	1.55263
$672$	0	0
$673$	36.7896	1.41813	0.709067	−	0.705141i	$-0.249116\pi$
$673$	36.7896	1.41813	0.709067	+	0.705141i	$0.249116\pi$
$674$	0	0
$675$	1.21037	0.0465872
$676$	0	0
$677$	31.4562	1.20896	0.604479	−	0.796621i	$-0.293381\pi$
$677$	31.4562	1.20896	0.604479	+	0.796621i	$0.293381\pi$
$678$	0	0
$679$	−55.6629	−2.13615
$680$	0	0
$681$	0.598699	0.0229422
$682$	0	0
$683$	−25.5292	−0.976847	−0.488424	−	0.872607i	$-0.662428\pi$
$683$	−25.5292	−0.976847	−0.488424	+	0.872607i	$0.662428\pi$
$684$	0	0
$685$	37.9059	1.44831
$686$	0	0
$687$	0.288204	0.0109957
$688$	0	0
$689$	17.3707	0.661769
$690$	0	0
$691$	21.0836	0.802059	0.401029	−	0.916065i	$-0.368653\pi$
$691$	21.0836	0.802059	0.401029	+	0.916065i	$0.368653\pi$
$692$	0	0
$693$	70.1064	2.66312
$694$	0	0
$695$	−2.86925	−0.108837
$696$	0	0
$697$	23.3653	0.885025
$698$	0	0
$699$	0.223219	0.00844292
$700$	0	0
$701$	35.9398	1.35743	0.678713	−	0.734403i	$-0.262538\pi$
$701$	35.9398	1.35743	0.678713	+	0.734403i	$0.262538\pi$
$702$	0	0
$703$	−6.56047	−0.247433
$704$	0	0
$705$	0.914746	0.0344513
$706$	0	0
$707$	31.2937	1.17692
$708$	0	0
$709$	26.2979	0.987639	0.493819	−	0.869565i	$-0.335600\pi$
$709$	26.2979	0.987639	0.493819	+	0.869565i	$0.335600\pi$
$710$	0	0
$711$	22.0661	0.827542
$712$	0	0
$713$	−9.35873	−0.350487
$714$	0	0
$715$	43.4070	1.62333
$716$	0	0
$717$	−0.205606	−0.00767851
$718$	0	0
$719$	43.9867	1.64043	0.820214	−	0.572056i	$-0.193854\pi$
$719$	43.9867	1.64043	0.820214	+	0.572056i	$0.193854\pi$
$720$	0	0
$721$	−80.6904	−3.00507
$722$	0	0
$723$	0.0507939	0.00188905
$724$	0	0
$725$	49.3087	1.83128
$726$	0	0
$727$	32.2403	1.19572	0.597862	−	0.801599i	$-0.296017\pi$
$727$	32.2403	1.19572	0.597862	+	0.801599i	$0.296017\pi$
$728$	0	0
$729$	−26.9680	−0.998816
$730$	0	0
$731$	−7.94394	−0.293817
$732$	0	0
$733$	−16.4841	−0.608853	−0.304427	−	0.952536i	$-0.598465\pi$
$733$	−16.4841	−0.608853	−0.304427	+	0.952536i	$0.598465\pi$
$734$	0	0
$735$	−1.42111	−0.0524184
$736$	0	0
$737$	18.6292	0.686217
$738$	0	0
$739$	7.81818	0.287596	0.143798	−	0.989607i	$-0.454068\pi$
$739$	7.81818	0.287596	0.143798	+	0.989607i	$0.454068\pi$
$740$	0	0
$741$	0.0595644	0.00218815
$742$	0	0
$743$	−17.5180	−0.642673	−0.321337	−	0.946965i	$-0.604132\pi$
$743$	−17.5180	−0.642673	−0.321337	+	0.946965i	$0.604132\pi$
$744$	0	0
$745$	29.0163	1.06307
$746$	0	0
$747$	−14.9225	−0.545987
$748$	0	0
$749$	−46.4204	−1.69617
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−0.216129	−0.00787618
$754$	0	0
$755$	37.6679	1.37088
$756$	0	0
$757$	−26.9809	−0.980638	−0.490319	−	0.871543i	$-0.663120\pi$
$757$	−26.9809	−0.980638	−0.490319	+	0.871543i	$0.663120\pi$
$758$	0	0
$759$	0.338698	0.0122939
$760$	0	0
$761$	−45.0869	−1.63440	−0.817200	−	0.576354i	$-0.804475\pi$
$761$	−45.0869	−1.63440	−0.817200	+	0.576354i	$0.804475\pi$
$762$	0	0
$763$	36.7005	1.32865
$764$	0	0
$765$	29.4319	1.06411
$766$	0	0
$767$	20.3984	0.736542
$768$	0	0
$769$	−32.8854	−1.18588	−0.592939	−	0.805247i	$-0.702032\pi$
$769$	−32.8854	−1.18588	−0.592939	+	0.805247i	$0.702032\pi$
$770$	0	0
$771$	−0.399792	−0.0143981
$772$	0	0
$773$	43.2064	1.55402	0.777012	−	0.629485i	$-0.216734\pi$
$773$	43.2064	1.55402	0.777012	+	0.629485i	$0.216734\pi$
$774$	0	0
$775$	27.1572	0.975515
$776$	0	0
$777$	0.764657	0.0274319
$778$	0	0
$779$	8.69515	0.311536
$780$	0	0
$781$	−60.0906	−2.15021
$782$	0	0
$783$	0.868130	0.0310244
$784$	0	0
$785$	34.4770	1.23054
$786$	0	0
$787$	−13.2567	−0.472550	−0.236275	−	0.971686i	$-0.575927\pi$
$787$	−13.2567	−0.472550	−0.236275	+	0.971686i	$0.575927\pi$
$788$	0	0
$789$	0.130378	0.00464159
$790$	0	0
$791$	−54.1175	−1.92420
$792$	0	0
$793$	20.1773	0.716515
$794$	0	0
$795$	−0.630492	−0.0223613
$796$	0	0
$797$	−7.43211	−0.263259	−0.131629	−	0.991299i	$-0.542021\pi$
$797$	−7.43211	−0.263259	−0.131629	+	0.991299i	$0.542021\pi$
$798$	0	0
$799$	27.7543	0.981876
$800$	0	0
$801$	−31.9880	−1.13024
$802$	0	0
$803$	−62.9009	−2.21973
$804$	0	0
$805$	49.9802	1.76157
$806$	0	0
$807$	−0.111779	−0.00393481
$808$	0	0
$809$	5.32161	0.187098	0.0935488	−	0.995615i	$-0.470179\pi$
$809$	5.32161	0.187098	0.0935488	+	0.995615i	$0.470179\pi$
$810$	0	0
$811$	−3.35103	−0.117671	−0.0588353	−	0.998268i	$-0.518739\pi$
$811$	−3.35103	−0.117671	−0.0588353	+	0.998268i	$0.518739\pi$
$812$	0	0
$813$	−0.0651326	−0.00228430
$814$	0	0
$815$	56.5940	1.98240
$816$	0	0
$817$	−2.95625	−0.103426
$818$	0	0
$819$	35.1715	1.22899
$820$	0	0
$821$	−6.52914	−0.227868	−0.113934	−	0.993488i	$-0.536345\pi$
$821$	−6.52914	−0.227868	−0.113934	+	0.993488i	$0.536345\pi$
$822$	0	0
$823$	−51.6467	−1.80029	−0.900145	−	0.435591i	$-0.856539\pi$
$823$	−51.6467	−1.80029	−0.900145	+	0.435591i	$0.856539\pi$
$824$	0	0
$825$	−0.982834	−0.0342179
$826$	0	0
$827$	−33.7898	−1.17499	−0.587493	−	0.809229i	$-0.699885\pi$
$827$	−33.7898	−1.17499	−0.587493	+	0.809229i	$0.699885\pi$
$828$	0	0
$829$	24.5924	0.854129	0.427065	−	0.904221i	$-0.359548\pi$
$829$	24.5924	0.854129	0.427065	+	0.904221i	$0.359548\pi$
$830$	0	0
$831$	−0.445128	−0.0154413
$832$	0	0
$833$	−43.1178	−1.49394
$834$	0	0
$835$	25.2467	0.873698
$836$	0	0
$837$	0.478130	0.0165266
$838$	0	0
$839$	35.8463	1.23755	0.618776	−	0.785567i	$-0.287629\pi$
$839$	35.8463	1.23755	0.618776	+	0.785567i	$0.287629\pi$
$840$	0	0
$841$	6.36634	0.219529
$842$	0	0
$843$	0.461400	0.0158915
$844$	0	0
$845$	−25.6179	−0.881282
$846$	0	0
$847$	−61.0885	−2.09903
$848$	0	0
$849$	−0.551043	−0.0189118
$850$	0	0
$851$	−18.7152	−0.641548
$852$	0	0
$853$	32.3633	1.10810	0.554049	−	0.832484i	$-0.313082\pi$
$853$	32.3633	1.10810	0.554049	+	0.832484i	$0.313082\pi$
$854$	0	0
$855$	10.9527	0.374576
$856$	0	0
$857$	24.0679	0.822144	0.411072	−	0.911603i	$-0.365155\pi$
$857$	24.0679	0.822144	0.411072	+	0.911603i	$0.365155\pi$
$858$	0	0
$859$	−44.5105	−1.51868	−0.759340	−	0.650695i	$-0.774478\pi$
$859$	−44.5105	−1.51868	−0.759340	+	0.650695i	$0.774478\pi$
$860$	0	0
$861$	−1.01346	−0.0345388
$862$	0	0
$863$	6.52826	0.222225	0.111112	−	0.993808i	$-0.464559\pi$
$863$	6.52826	0.222225	0.111112	+	0.993808i	$0.464559\pi$
$864$	0	0
$865$	83.7713	2.84831
$866$	0	0
$867$	0.237379	0.00806182
$868$	0	0
$869$	−35.8393	−1.21576
$870$	0	0
$871$	9.34606	0.316679
$872$	0	0
$873$	−34.7976	−1.17772
$874$	0	0
$875$	−57.5732	−1.94633
$876$	0	0
$877$	−47.8646	−1.61627	−0.808136	−	0.588996i	$-0.799523\pi$
$877$	−47.8646	−1.61627	−0.808136	+	0.588996i	$0.799523\pi$
$878$	0	0
$879$	−0.404759	−0.0136522
$880$	0	0
$881$	−36.0735	−1.21535	−0.607674	−	0.794187i	$-0.707897\pi$
$881$	−36.0735	−1.21535	−0.607674	+	0.794187i	$0.707897\pi$
$882$	0	0
$883$	−41.6778	−1.40257	−0.701285	−	0.712881i	$-0.747390\pi$
$883$	−41.6778	−1.40257	−0.701285	+	0.712881i	$0.747390\pi$
$884$	0	0
$885$	−0.740387	−0.0248878
$886$	0	0
$887$	−1.10519	−0.0371086	−0.0185543	−	0.999828i	$-0.505906\pi$
$887$	−1.10519	−0.0371086	−0.0185543	+	0.999828i	$0.505906\pi$
$888$	0	0
$889$	9.16940	0.307532
$890$	0	0
$891$	43.8183	1.46797
$892$	0	0
$893$	10.3284	0.345628
$894$	0	0
$895$	−50.1138	−1.67512
$896$	0	0
$897$	0.169920	0.00567348
$898$	0	0
$899$	19.4783	0.649638
$900$	0	0
$901$	−19.1297	−0.637304
$902$	0	0
$903$	0.344566	0.0114665
$904$	0	0
$905$	5.73090	0.190502
$906$	0	0
$907$	11.9847	0.397944	0.198972	−	0.980005i	$-0.436240\pi$
$907$	11.9847	0.397944	0.198972	+	0.980005i	$0.436240\pi$
$908$	0	0
$909$	19.5632	0.648872
$910$	0	0
$911$	−43.6363	−1.44574	−0.722868	−	0.690986i	$-0.757177\pi$
$911$	−43.6363	−1.44574	−0.722868	+	0.690986i	$0.757177\pi$
$912$	0	0
$913$	24.2369	0.802124
$914$	0	0
$915$	−0.732362	−0.0242111
$916$	0	0
$917$	62.3551	2.05915
$918$	0	0
$919$	−11.3742	−0.375200	−0.187600	−	0.982245i	$-0.560071\pi$
$919$	−11.3742	−0.375200	−0.187600	+	0.982245i	$0.560071\pi$
$920$	0	0
$921$	0.258826	0.00852861
$922$	0	0
$923$	−30.1467	−0.992291
$924$	0	0
$925$	54.3078	1.78563
$926$	0	0
$927$	−50.4435	−1.65678
$928$	0	0
$929$	−15.5039	−0.508666	−0.254333	−	0.967117i	$-0.581856\pi$
$929$	−15.5039	−0.508666	−0.254333	+	0.967117i	$0.581856\pi$
$930$	0	0
$931$	−16.0458	−0.525881
$932$	0	0
$933$	0.288665	0.00945048
$934$	0	0
$935$	−47.8027	−1.56331
$936$	0	0
$937$	−33.9650	−1.10959	−0.554794	−	0.831988i	$-0.687203\pi$
$937$	−33.9650	−1.10959	−0.554794	+	0.831988i	$0.687203\pi$
$938$	0	0
$939$	0.0544810	0.00177792
$940$	0	0
$941$	−35.6611	−1.16252	−0.581260	−	0.813718i	$-0.697440\pi$
$941$	−35.6611	−1.16252	−0.581260	+	0.813718i	$0.697440\pi$
$942$	0	0
$943$	24.8048	0.807756
$944$	0	0
$945$	−2.55345	−0.0830637
$946$	0	0
$947$	−18.6041	−0.604552	−0.302276	−	0.953220i	$-0.597746\pi$
$947$	−18.6041	−0.604552	−0.302276	+	0.953220i	$0.597746\pi$
$948$	0	0
$949$	−31.5566	−1.02437
$950$	0	0
$951$	0.148987	0.00483122
$952$	0	0
$953$	−54.6187	−1.76927	−0.884637	−	0.466280i	$-0.845594\pi$
$953$	−54.6187	−1.76927	−0.884637	+	0.466280i	$0.845594\pi$
$954$	0	0
$955$	9.79311	0.316898
$956$	0	0
$957$	−0.704931	−0.0227872
$958$	0	0
$959$	−49.8853	−1.61088
$960$	0	0
$961$	−20.2722	−0.653941
$962$	0	0
$963$	−29.0197	−0.935147
$964$	0	0
$965$	85.5951	2.75540
$966$	0	0
$967$	41.9213	1.34810	0.674049	−	0.738687i	$-0.264554\pi$
$967$	41.9213	1.34810	0.674049	+	0.738687i	$0.264554\pi$
$968$	0	0
$969$	−0.0655963	−0.00210726
$970$	0	0
$971$	−20.4238	−0.655432	−0.327716	−	0.944776i	$-0.606279\pi$
$971$	−20.4238	−0.655432	−0.327716	+	0.944776i	$0.606279\pi$
$972$	0	0
$973$	3.77603	0.121054
$974$	0	0
$975$	−0.493076	−0.0157911
$976$	0	0
$977$	−15.6782	−0.501590	−0.250795	−	0.968040i	$-0.580692\pi$
$977$	−15.6782	−0.501590	−0.250795	+	0.968040i	$0.580692\pi$
$978$	0	0
$979$	51.9543	1.66047
$980$	0	0
$981$	22.9433	0.732522
$982$	0	0
$983$	37.4028	1.19296	0.596481	−	0.802627i	$-0.296565\pi$
$983$	37.4028	1.19296	0.596481	+	0.802627i	$0.296565\pi$
$984$	0	0
$985$	95.1185	3.03073
$986$	0	0
$987$	−1.20383	−0.0383185
$988$	0	0
$989$	−8.43335	−0.268165
$990$	0	0
$991$	23.9275	0.760082	0.380041	−	0.924970i	$-0.375910\pi$
$991$	23.9275	0.760082	0.380041	+	0.924970i	$0.375910\pi$
$992$	0	0
$993$	−0.0325277	−0.00103224
$994$	0	0
$995$	−42.1461	−1.33612
$996$	0	0
$997$	−41.9335	−1.32805	−0.664024	−	0.747711i	$-0.731153\pi$
$997$	−41.9335	−1.32805	−0.664024	+	0.747711i	$0.731153\pi$
$998$	0	0
$999$	0.956143	0.0302510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.25	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.25	✓	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)

\(f(q)\)	\(=\)	\(q-0.0243322 q^{3} +3.64574 q^{5} -4.79791 q^{7} -2.99941 q^{9} +O(q^{10})\)	\(q-0.0243322 q^{3} +3.64574 q^{5} -4.79791 q^{7} -2.99941 q^{9} +4.87158 q^{11} +2.44401 q^{13} -0.0887089 q^{15} -2.69151 q^{17} -1.00162 q^{19} +0.116744 q^{21} -2.85733 q^{23} +8.29142 q^{25} +0.145979 q^{27} +5.94696 q^{29} +3.27534 q^{31} -0.118536 q^{33} -17.4919 q^{35} +6.54988 q^{37} -0.0594682 q^{39} -8.68111 q^{41} +2.95148 q^{43} -10.9351 q^{45} -10.3118 q^{47} +16.0199 q^{49} +0.0654904 q^{51} +7.10743 q^{53} +17.7605 q^{55} +0.0243715 q^{57} +8.34625 q^{59} +8.25579 q^{61} +14.3909 q^{63} +8.91024 q^{65} +3.82406 q^{67} +0.0695252 q^{69} -12.3349 q^{71} -12.9118 q^{73} -0.201748 q^{75} -23.3734 q^{77} -7.35680 q^{79} +8.99467 q^{81} +4.97516 q^{83} -9.81256 q^{85} -0.144703 q^{87} +10.6648 q^{89} -11.7262 q^{91} -0.0796962 q^{93} -3.65163 q^{95} +11.6015 q^{97} -14.6119 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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

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Database

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