LMFDB - Embedded newform 6008.2.a.e.1.41

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			1.41
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)

$f(q)$	$=$	$q+2.21288 q^{3} +0.631281 q^{5} +4.31774 q^{7} +1.89684 q^{9} +O(q^{10})$	$q+2.21288 q^{3} +0.631281 q^{5} +4.31774 q^{7} +1.89684 q^{9} -2.74984 q^{11} -3.87462 q^{13} +1.39695 q^{15} +3.58469 q^{17} +4.36126 q^{19} +9.55464 q^{21} -4.21696 q^{23} -4.60148 q^{25} -2.44116 q^{27} +4.85071 q^{29} +7.80799 q^{31} -6.08508 q^{33} +2.72570 q^{35} -2.92265 q^{37} -8.57408 q^{39} +2.10427 q^{41} +6.74428 q^{43} +1.19744 q^{45} +3.69909 q^{47} +11.6429 q^{49} +7.93249 q^{51} -3.10581 q^{53} -1.73592 q^{55} +9.65095 q^{57} +3.55015 q^{59} +14.8343 q^{61} +8.19005 q^{63} -2.44598 q^{65} +13.5276 q^{67} -9.33162 q^{69} +5.91516 q^{71} +10.0555 q^{73} -10.1825 q^{75} -11.8731 q^{77} +3.57721 q^{79} -11.0925 q^{81} -1.16410 q^{83} +2.26295 q^{85} +10.7340 q^{87} +1.16450 q^{89} -16.7296 q^{91} +17.2782 q^{93} +2.75318 q^{95} -12.5312 q^{97} -5.21601 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.21288	1.27761	0.638803	−	0.769370i	$-0.279430\pi$
$3$	2.21288	1.27761	0.638803	+	0.769370i	$0.279430\pi$
$4$	0	0
$5$	0.631281	0.282317	0.141159	−	0.989987i	$-0.454917\pi$
$5$	0.631281	0.282317	0.141159	+	0.989987i	$0.454917\pi$
$6$	0	0
$7$	4.31774	1.63195	0.815976	−	0.578086i	$-0.196200\pi$
$7$	4.31774	1.63195	0.815976	+	0.578086i	$0.196200\pi$
$8$	0	0
$9$	1.89684	0.632280
$10$	0	0
$11$	−2.74984	−0.829109	−0.414555	−	0.910024i	$-0.636063\pi$
$11$	−2.74984	−0.829109	−0.414555	+	0.910024i	$0.636063\pi$
$12$	0	0
$13$	−3.87462	−1.07463	−0.537314	−	0.843382i	$-0.680561\pi$
$13$	−3.87462	−1.07463	−0.537314	+	0.843382i	$0.680561\pi$
$14$	0	0
$15$	1.39695	0.360691
$16$	0	0
$17$	3.58469	0.869415	0.434707	−	0.900572i	$-0.356852\pi$
$17$	3.58469	0.869415	0.434707	+	0.900572i	$0.356852\pi$
$18$	0	0
$19$	4.36126	1.00054	0.500271	−	0.865869i	$-0.333234\pi$
$19$	4.36126	1.00054	0.500271	+	0.865869i	$0.333234\pi$
$20$	0	0
$21$	9.55464	2.08499
$22$	0	0
$23$	−4.21696	−0.879296	−0.439648	−	0.898170i	$-0.644897\pi$
$23$	−4.21696	−0.879296	−0.439648	+	0.898170i	$0.644897\pi$
$24$	0	0
$25$	−4.60148	−0.920297
$26$	0	0
$27$	−2.44116	−0.469802
$28$	0	0
$29$	4.85071	0.900753	0.450377	−	0.892839i	$-0.351290\pi$
$29$	4.85071	0.900753	0.450377	+	0.892839i	$0.351290\pi$
$30$	0	0
$31$	7.80799	1.40236	0.701179	−	0.712986i	$-0.252658\pi$
$31$	7.80799	1.40236	0.701179	+	0.712986i	$0.252658\pi$
$32$	0	0
$33$	−6.08508	−1.05928
$34$	0	0
$35$	2.72570	0.460728
$36$	0	0
$37$	−2.92265	−0.480481	−0.240240	−	0.970713i	$-0.577226\pi$
$37$	−2.92265	−0.480481	−0.240240	+	0.970713i	$0.577226\pi$
$38$	0	0
$39$	−8.57408	−1.37295
$40$	0	0
$41$	2.10427	0.328631	0.164316	−	0.986408i	$-0.447458\pi$
$41$	2.10427	0.328631	0.164316	+	0.986408i	$0.447458\pi$
$42$	0	0
$43$	6.74428	1.02849	0.514247	−	0.857642i	$-0.328072\pi$
$43$	6.74428	1.02849	0.514247	+	0.857642i	$0.328072\pi$
$44$	0	0
$45$	1.19744	0.178503
$46$	0	0
$47$	3.69909	0.539567	0.269784	−	0.962921i	$-0.413048\pi$
$47$	3.69909	0.539567	0.269784	+	0.962921i	$0.413048\pi$
$48$	0	0
$49$	11.6429	1.66327
$50$	0	0
$51$	7.93249	1.11077
$52$	0	0
$53$	−3.10581	−0.426616	−0.213308	−	0.976985i	$-0.568424\pi$
$53$	−3.10581	−0.426616	−0.213308	+	0.976985i	$0.568424\pi$
$54$	0	0
$55$	−1.73592	−0.234072
$56$	0	0
$57$	9.65095	1.27830
$58$	0	0
$59$	3.55015	0.462191	0.231095	−	0.972931i	$-0.425769\pi$
$59$	3.55015	0.462191	0.231095	+	0.972931i	$0.425769\pi$
$60$	0	0
$61$	14.8343	1.89934	0.949671	−	0.313249i	$-0.101417\pi$
$61$	14.8343	1.89934	0.949671	+	0.313249i	$0.101417\pi$
$62$	0	0
$63$	8.19005	1.03185
$64$	0	0
$65$	−2.44598	−0.303386
$66$	0	0
$67$	13.5276	1.65266	0.826328	−	0.563189i	$-0.190426\pi$
$67$	13.5276	1.65266	0.826328	+	0.563189i	$0.190426\pi$
$68$	0	0
$69$	−9.33162	−1.12339
$70$	0	0
$71$	5.91516	0.702001	0.351000	−	0.936375i	$-0.385842\pi$
$71$	5.91516	0.702001	0.351000	+	0.936375i	$0.385842\pi$
$72$	0	0
$73$	10.0555	1.17690	0.588451	−	0.808533i	$-0.299738\pi$
$73$	10.0555	1.17690	0.588451	+	0.808533i	$0.299738\pi$
$74$	0	0
$75$	−10.1825	−1.17578
$76$	0	0
$77$	−11.8731	−1.35307
$78$	0	0
$79$	3.57721	0.402467	0.201234	−	0.979543i	$-0.435505\pi$
$79$	3.57721	0.402467	0.201234	+	0.979543i	$0.435505\pi$
$80$	0	0
$81$	−11.0925	−1.23250
$82$	0	0
$83$	−1.16410	−0.127776	−0.0638882	−	0.997957i	$-0.520350\pi$
$83$	−1.16410	−0.127776	−0.0638882	+	0.997957i	$0.520350\pi$
$84$	0	0
$85$	2.26295	0.245451
$86$	0	0
$87$	10.7340	1.15081
$88$	0	0
$89$	1.16450	0.123437	0.0617183	−	0.998094i	$-0.480342\pi$
$89$	1.16450	0.123437	0.0617183	+	0.998094i	$0.480342\pi$
$90$	0	0
$91$	−16.7296	−1.75374
$92$	0	0
$93$	17.2782	1.79166
$94$	0	0
$95$	2.75318	0.282470
$96$	0	0
$97$	−12.5312	−1.27235	−0.636173	−	0.771546i	$-0.719484\pi$
$97$	−12.5312	−1.27235	−0.636173	+	0.771546i	$0.719484\pi$
$98$	0	0
$99$	−5.21601	−0.524229
$100$	0	0
$101$	−9.26514	−0.921916	−0.460958	−	0.887422i	$-0.652494\pi$
$101$	−9.26514	−0.921916	−0.460958	+	0.887422i	$0.652494\pi$
$102$	0	0
$103$	−1.27125	−0.125260	−0.0626300	−	0.998037i	$-0.519949\pi$
$103$	−1.27125	−0.125260	−0.0626300	+	0.998037i	$0.519949\pi$
$104$	0	0
$105$	6.03166	0.588629
$106$	0	0
$107$	6.95788	0.672644	0.336322	−	0.941747i	$-0.390817\pi$
$107$	6.95788	0.672644	0.336322	+	0.941747i	$0.390817\pi$
$108$	0	0
$109$	0.584876	0.0560209	0.0280105	−	0.999608i	$-0.491083\pi$
$109$	0.584876	0.0560209	0.0280105	+	0.999608i	$0.491083\pi$
$110$	0	0
$111$	−6.46748	−0.613866
$112$	0	0
$113$	−7.48539	−0.704167	−0.352083	−	0.935969i	$-0.614527\pi$
$113$	−7.48539	−0.704167	−0.352083	+	0.935969i	$0.614527\pi$
$114$	0	0
$115$	−2.66208	−0.248240
$116$	0	0
$117$	−7.34954	−0.679465
$118$	0	0
$119$	15.4778	1.41884
$120$	0	0
$121$	−3.43836	−0.312578
$122$	0	0
$123$	4.65649	0.419862
$124$	0	0
$125$	−6.06123	−0.542133
$126$	0	0
$127$	8.11738	0.720301	0.360151	−	0.932894i	$-0.382725\pi$
$127$	8.11738	0.720301	0.360151	+	0.932894i	$0.382725\pi$
$128$	0	0
$129$	14.9243	1.31401
$130$	0	0
$131$	9.04443	0.790216	0.395108	−	0.918635i	$-0.370707\pi$
$131$	9.04443	0.790216	0.395108	+	0.918635i	$0.370707\pi$
$132$	0	0
$133$	18.8308	1.63284
$134$	0	0
$135$	−1.54106	−0.132633
$136$	0	0
$137$	9.13433	0.780398	0.390199	−	0.920730i	$-0.372406\pi$
$137$	9.13433	0.780398	0.390199	+	0.920730i	$0.372406\pi$
$138$	0	0
$139$	9.67848	0.820918	0.410459	−	0.911879i	$-0.365369\pi$
$139$	9.67848	0.820918	0.410459	+	0.911879i	$0.365369\pi$
$140$	0	0
$141$	8.18564	0.689355
$142$	0	0
$143$	10.6546	0.890984
$144$	0	0
$145$	3.06216	0.254298
$146$	0	0
$147$	25.7643	2.12500
$148$	0	0
$149$	3.89320	0.318944	0.159472	−	0.987202i	$-0.449021\pi$
$149$	3.89320	0.318944	0.159472	+	0.987202i	$0.449021\pi$
$150$	0	0
$151$	−8.83702	−0.719146	−0.359573	−	0.933117i	$-0.617078\pi$
$151$	−8.83702	−0.719146	−0.359573	+	0.933117i	$0.617078\pi$
$152$	0	0
$153$	6.79958	0.549713
$154$	0	0
$155$	4.92904	0.395910
$156$	0	0
$157$	11.4867	0.916736	0.458368	−	0.888762i	$-0.348434\pi$
$157$	11.4867	0.916736	0.458368	+	0.888762i	$0.348434\pi$
$158$	0	0
$159$	−6.87278	−0.545047
$160$	0	0
$161$	−18.2077	−1.43497
$162$	0	0
$163$	−18.0117	−1.41078	−0.705392	−	0.708818i	$-0.749229\pi$
$163$	−18.0117	−1.41078	−0.705392	+	0.708818i	$0.749229\pi$
$164$	0	0
$165$	−3.84139	−0.299052
$166$	0	0
$167$	8.30351	0.642545	0.321273	−	0.946987i	$-0.395889\pi$
$167$	8.30351	0.642545	0.321273	+	0.946987i	$0.395889\pi$
$168$	0	0
$169$	2.01272	0.154824
$170$	0	0
$171$	8.27261	0.632622
$172$	0	0
$173$	−16.4024	−1.24705	−0.623527	−	0.781802i	$-0.714301\pi$
$173$	−16.4024	−1.24705	−0.623527	+	0.781802i	$0.714301\pi$
$174$	0	0
$175$	−19.8680	−1.50188
$176$	0	0
$177$	7.85607	0.590498
$178$	0	0
$179$	−6.43860	−0.481243	−0.240622	−	0.970619i	$-0.577351\pi$
$179$	−6.43860	−0.481243	−0.240622	+	0.970619i	$0.577351\pi$
$180$	0	0
$181$	9.27665	0.689528	0.344764	−	0.938689i	$-0.387959\pi$
$181$	9.27665	0.689528	0.344764	+	0.938689i	$0.387959\pi$
$182$	0	0
$183$	32.8266	2.42661
$184$	0	0
$185$	−1.84501	−0.135648
$186$	0	0
$187$	−9.85734	−0.720840
$188$	0	0
$189$	−10.5403	−0.766694
$190$	0	0
$191$	2.88178	0.208518	0.104259	−	0.994550i	$-0.466753\pi$
$191$	2.88178	0.208518	0.104259	+	0.994550i	$0.466753\pi$
$192$	0	0
$193$	−23.9742	−1.72570	−0.862849	−	0.505462i	$-0.831322\pi$
$193$	−23.9742	−1.72570	−0.862849	+	0.505462i	$0.831322\pi$
$194$	0	0
$195$	−5.41265	−0.387608
$196$	0	0
$197$	0.980826	0.0698810	0.0349405	−	0.999389i	$-0.488876\pi$
$197$	0.980826	0.0698810	0.0349405	+	0.999389i	$0.488876\pi$
$198$	0	0
$199$	−18.3854	−1.30331	−0.651653	−	0.758517i	$-0.725924\pi$
$199$	−18.3854	−1.30331	−0.651653	+	0.758517i	$0.725924\pi$
$200$	0	0
$201$	29.9349	2.11145
$202$	0	0
$203$	20.9441	1.46999
$204$	0	0
$205$	1.32838	0.0927783
$206$	0	0
$207$	−7.99888	−0.555961
$208$	0	0
$209$	−11.9928	−0.829559
$210$	0	0
$211$	7.91434	0.544846	0.272423	−	0.962178i	$-0.412175\pi$
$211$	7.91434	0.544846	0.272423	+	0.962178i	$0.412175\pi$
$212$	0	0
$213$	13.0896	0.896881
$214$	0	0
$215$	4.25754	0.290362
$216$	0	0
$217$	33.7129	2.28858
$218$	0	0
$219$	22.2515	1.50362
$220$	0	0
$221$	−13.8893	−0.934297
$222$	0	0
$223$	−11.0381	−0.739164	−0.369582	−	0.929198i	$-0.620499\pi$
$223$	−11.0381	−0.739164	−0.369582	+	0.929198i	$0.620499\pi$
$224$	0	0
$225$	−8.72827	−0.581885
$226$	0	0
$227$	12.4984	0.829548	0.414774	−	0.909924i	$-0.363861\pi$
$227$	12.4984	0.829548	0.414774	+	0.909924i	$0.363861\pi$
$228$	0	0
$229$	−28.8715	−1.90788	−0.953941	−	0.299993i	$-0.903016\pi$
$229$	−28.8715	−1.90788	−0.953941	+	0.299993i	$0.903016\pi$
$230$	0	0
$231$	−26.2738	−1.72869
$232$	0	0
$233$	−23.1048	−1.51364	−0.756821	−	0.653622i	$-0.773249\pi$
$233$	−23.1048	−1.51364	−0.756821	+	0.653622i	$0.773249\pi$
$234$	0	0
$235$	2.33516	0.152329
$236$	0	0
$237$	7.91593	0.514195
$238$	0	0
$239$	−16.0749	−1.03980	−0.519899	−	0.854228i	$-0.674031\pi$
$239$	−16.0749	−1.03980	−0.519899	+	0.854228i	$0.674031\pi$
$240$	0	0
$241$	3.20222	0.206273	0.103137	−	0.994667i	$-0.467112\pi$
$241$	3.20222	0.206273	0.103137	+	0.994667i	$0.467112\pi$
$242$	0	0
$243$	−17.2229	−1.10485
$244$	0	0
$245$	7.34991	0.469569
$246$	0	0
$247$	−16.8983	−1.07521
$248$	0	0
$249$	−2.57601	−0.163248
$250$	0	0
$251$	−23.1396	−1.46056	−0.730278	−	0.683150i	$-0.760610\pi$
$251$	−23.1396	−1.46056	−0.730278	+	0.683150i	$0.760610\pi$
$252$	0	0
$253$	11.5960	0.729032
$254$	0	0
$255$	5.00763	0.313590
$256$	0	0
$257$	−3.47729	−0.216907	−0.108454	−	0.994102i	$-0.534590\pi$
$257$	−3.47729	−0.216907	−0.108454	+	0.994102i	$0.534590\pi$
$258$	0	0
$259$	−12.6192	−0.784121
$260$	0	0
$261$	9.20101	0.569528
$262$	0	0
$263$	−16.2698	−1.00324	−0.501619	−	0.865089i	$-0.667262\pi$
$263$	−16.2698	−1.00324	−0.501619	+	0.865089i	$0.667262\pi$
$264$	0	0
$265$	−1.96064	−0.120441
$266$	0	0
$267$	2.57690	0.157703
$268$	0	0
$269$	−4.20248	−0.256230	−0.128115	−	0.991759i	$-0.540893\pi$
$269$	−4.20248	−0.256230	−0.128115	+	0.991759i	$0.540893\pi$
$270$	0	0
$271$	−10.1689	−0.617714	−0.308857	−	0.951109i	$-0.599946\pi$
$271$	−10.1689	−0.617714	−0.308857	+	0.951109i	$0.599946\pi$
$272$	0	0
$273$	−37.0206	−2.24059
$274$	0	0
$275$	12.6534	0.763027
$276$	0	0
$277$	−0.162834	−0.00978374	−0.00489187	−	0.999988i	$-0.501557\pi$
$277$	−0.162834	−0.00978374	−0.00489187	+	0.999988i	$0.501557\pi$
$278$	0	0
$279$	14.8105	0.886682
$280$	0	0
$281$	−9.64772	−0.575534	−0.287767	−	0.957700i	$-0.592913\pi$
$281$	−9.64772	−0.575534	−0.287767	+	0.957700i	$0.592913\pi$
$282$	0	0
$283$	13.9268	0.827865	0.413932	−	0.910308i	$-0.364155\pi$
$283$	13.9268	0.827865	0.413932	+	0.910308i	$0.364155\pi$
$284$	0	0
$285$	6.09246	0.360886
$286$	0	0
$287$	9.08567	0.536310
$288$	0	0
$289$	−4.15000	−0.244118
$290$	0	0
$291$	−27.7300	−1.62556
$292$	0	0
$293$	−6.12245	−0.357677	−0.178839	−	0.983878i	$-0.557234\pi$
$293$	−6.12245	−0.357677	−0.178839	+	0.983878i	$0.557234\pi$
$294$	0	0
$295$	2.24114	0.130484
$296$	0	0
$297$	6.71282	0.389517
$298$	0	0
$299$	16.3391	0.944916
$300$	0	0
$301$	29.1201	1.67845
$302$	0	0
$303$	−20.5026	−1.17785
$304$	0	0
$305$	9.36463	0.536217
$306$	0	0
$307$	18.8744	1.07722	0.538611	−	0.842555i	$-0.318949\pi$
$307$	18.8744	1.07722	0.538611	+	0.842555i	$0.318949\pi$
$308$	0	0
$309$	−2.81313	−0.160033
$310$	0	0
$311$	−13.3670	−0.757976	−0.378988	−	0.925402i	$-0.623728\pi$
$311$	−13.3670	−0.757976	−0.378988	+	0.925402i	$0.623728\pi$
$312$	0	0
$313$	7.22783	0.408541	0.204270	−	0.978914i	$-0.434518\pi$
$313$	7.22783	0.408541	0.204270	+	0.978914i	$0.434518\pi$
$314$	0	0
$315$	5.17022	0.291309
$316$	0	0
$317$	12.4311	0.698199	0.349099	−	0.937086i	$-0.386488\pi$
$317$	12.4311	0.698199	0.349099	+	0.937086i	$0.386488\pi$
$318$	0	0
$319$	−13.3387	−0.746823
$320$	0	0
$321$	15.3970	0.859374
$322$	0	0
$323$	15.6338	0.869886
$324$	0	0
$325$	17.8290	0.988976
$326$	0	0
$327$	1.29426	0.0715727
$328$	0	0
$329$	15.9717	0.880548
$330$	0	0
$331$	−20.0366	−1.10131	−0.550657	−	0.834732i	$-0.685623\pi$
$331$	−20.0366	−1.10131	−0.550657	+	0.834732i	$0.685623\pi$
$332$	0	0
$333$	−5.54380	−0.303798
$334$	0	0
$335$	8.53970	0.466573
$336$	0	0
$337$	1.82672	0.0995077	0.0497539	−	0.998762i	$-0.484156\pi$
$337$	1.82672	0.0995077	0.0497539	+	0.998762i	$0.484156\pi$
$338$	0	0
$339$	−16.5643	−0.899648
$340$	0	0
$341$	−21.4708	−1.16271
$342$	0	0
$343$	20.0467	1.08242
$344$	0	0
$345$	−5.89087	−0.317154
$346$	0	0
$347$	−8.77305	−0.470962	−0.235481	−	0.971879i	$-0.575667\pi$
$347$	−8.77305	−0.470962	−0.235481	+	0.971879i	$0.575667\pi$
$348$	0	0
$349$	15.7589	0.843552	0.421776	−	0.906700i	$-0.361407\pi$
$349$	15.7589	0.843552	0.421776	+	0.906700i	$0.361407\pi$
$350$	0	0
$351$	9.45859	0.504862
$352$	0	0
$353$	−25.6824	−1.36693	−0.683467	−	0.729982i	$-0.739529\pi$
$353$	−25.6824	−1.36693	−0.683467	+	0.729982i	$0.739529\pi$
$354$	0	0
$355$	3.73413	0.198187
$356$	0	0
$357$	34.2504	1.81272
$358$	0	0
$359$	−28.4542	−1.50176	−0.750878	−	0.660441i	$-0.770369\pi$
$359$	−28.4542	−1.50176	−0.750878	+	0.660441i	$0.770369\pi$
$360$	0	0
$361$	0.0206039	0.00108442
$362$	0	0
$363$	−7.60867	−0.399352
$364$	0	0
$365$	6.34782	0.332260
$366$	0	0
$367$	−10.0337	−0.523753	−0.261876	−	0.965101i	$-0.584341\pi$
$367$	−10.0337	−0.523753	−0.261876	+	0.965101i	$0.584341\pi$
$368$	0	0
$369$	3.99145	0.207787
$370$	0	0
$371$	−13.4101	−0.696216
$372$	0	0
$373$	33.1622	1.71708	0.858538	−	0.512750i	$-0.171373\pi$
$373$	33.1622	1.71708	0.858538	+	0.512750i	$0.171373\pi$
$374$	0	0
$375$	−13.4128	−0.692633
$376$	0	0
$377$	−18.7947	−0.967975
$378$	0	0
$379$	−24.1456	−1.24028	−0.620138	−	0.784493i	$-0.712923\pi$
$379$	−24.1456	−1.24028	−0.620138	+	0.784493i	$0.712923\pi$
$380$	0	0
$381$	17.9628	0.920262
$382$	0	0
$383$	−14.7826	−0.755356	−0.377678	−	0.925937i	$-0.623277\pi$
$383$	−14.7826	−0.755356	−0.377678	+	0.925937i	$0.623277\pi$
$384$	0	0
$385$	−7.49526	−0.381994
$386$	0	0
$387$	12.7928	0.650296
$388$	0	0
$389$	31.7155	1.60804	0.804021	−	0.594601i	$-0.202690\pi$
$389$	31.7155	1.60804	0.804021	+	0.594601i	$0.202690\pi$
$390$	0	0
$391$	−15.1165	−0.764473
$392$	0	0
$393$	20.0142	1.00958
$394$	0	0
$395$	2.25822	0.113623
$396$	0	0
$397$	−7.28802	−0.365775	−0.182888	−	0.983134i	$-0.558544\pi$
$397$	−7.28802	−0.365775	−0.182888	+	0.983134i	$0.558544\pi$
$398$	0	0
$399$	41.6703	2.08612
$400$	0	0
$401$	−29.4462	−1.47047	−0.735237	−	0.677810i	$-0.762929\pi$
$401$	−29.4462	−1.47047	−0.735237	+	0.677810i	$0.762929\pi$
$402$	0	0
$403$	−30.2530	−1.50701
$404$	0	0
$405$	−7.00249	−0.347957
$406$	0	0
$407$	8.03683	0.398371
$408$	0	0
$409$	−2.97949	−0.147326	−0.0736631	−	0.997283i	$-0.523469\pi$
$409$	−2.97949	−0.147326	−0.0736631	+	0.997283i	$0.523469\pi$
$410$	0	0
$411$	20.2132	0.997042
$412$	0	0
$413$	15.3286	0.754273
$414$	0	0
$415$	−0.734873	−0.0360735
$416$	0	0
$417$	21.4173	1.04881
$418$	0	0
$419$	−31.9364	−1.56020	−0.780098	−	0.625657i	$-0.784831\pi$
$419$	−31.9364	−1.56020	−0.780098	+	0.625657i	$0.784831\pi$
$420$	0	0
$421$	1.36468	0.0665104	0.0332552	−	0.999447i	$-0.489413\pi$
$421$	1.36468	0.0665104	0.0332552	+	0.999447i	$0.489413\pi$
$422$	0	0
$423$	7.01657	0.341157
$424$	0	0
$425$	−16.4949	−0.800120
$426$	0	0
$427$	64.0508	3.09963
$428$	0	0
$429$	23.5774	1.13833
$430$	0	0
$431$	−19.0647	−0.918312	−0.459156	−	0.888356i	$-0.651848\pi$
$431$	−19.0647	−0.918312	−0.459156	+	0.888356i	$0.651848\pi$
$432$	0	0
$433$	−12.1700	−0.584851	−0.292425	−	0.956288i	$-0.594462\pi$
$433$	−12.1700	−0.584851	−0.292425	+	0.956288i	$0.594462\pi$
$434$	0	0
$435$	6.77619	0.324893
$436$	0	0
$437$	−18.3912	−0.879773
$438$	0	0
$439$	6.34306	0.302738	0.151369	−	0.988477i	$-0.451632\pi$
$439$	6.34306	0.302738	0.151369	+	0.988477i	$0.451632\pi$
$440$	0	0
$441$	22.0846	1.05165
$442$	0	0
$443$	35.7721	1.69958	0.849792	−	0.527119i	$-0.176728\pi$
$443$	35.7721	1.69958	0.849792	+	0.527119i	$0.176728\pi$
$444$	0	0
$445$	0.735126	0.0348483
$446$	0	0
$447$	8.61519	0.407484
$448$	0	0
$449$	−6.74464	−0.318299	−0.159150	−	0.987254i	$-0.550875\pi$
$449$	−6.74464	−0.318299	−0.159150	+	0.987254i	$0.550875\pi$
$450$	0	0
$451$	−5.78641	−0.272471
$452$	0	0
$453$	−19.5553	−0.918786
$454$	0	0
$455$	−10.5611	−0.495111
$456$	0	0
$457$	32.0403	1.49878	0.749390	−	0.662129i	$-0.230347\pi$
$457$	32.0403	1.49878	0.749390	+	0.662129i	$0.230347\pi$
$458$	0	0
$459$	−8.75081	−0.408453
$460$	0	0
$461$	5.61403	0.261471	0.130736	−	0.991417i	$-0.458266\pi$
$461$	5.61403	0.261471	0.130736	+	0.991417i	$0.458266\pi$
$462$	0	0
$463$	1.00691	0.0467953	0.0233976	−	0.999726i	$-0.492552\pi$
$463$	1.00691	0.0467953	0.0233976	+	0.999726i	$0.492552\pi$
$464$	0	0
$465$	10.9074	0.505817
$466$	0	0
$467$	−6.76760	−0.313167	−0.156584	−	0.987665i	$-0.550048\pi$
$467$	−6.76760	−0.313167	−0.156584	+	0.987665i	$0.550048\pi$
$468$	0	0
$469$	58.4085	2.69705
$470$	0	0
$471$	25.4186	1.17123
$472$	0	0
$473$	−18.5457	−0.852734
$474$	0	0
$475$	−20.0683	−0.920796
$476$	0	0
$477$	−5.89122	−0.269740
$478$	0	0
$479$	24.6305	1.12539	0.562697	−	0.826663i	$-0.309764\pi$
$479$	24.6305	1.12539	0.562697	+	0.826663i	$0.309764\pi$
$480$	0	0
$481$	11.3242	0.516338
$482$	0	0
$483$	−40.2915	−1.83333
$484$	0	0
$485$	−7.91068	−0.359206
$486$	0	0
$487$	28.6500	1.29826	0.649128	−	0.760680i	$-0.275134\pi$
$487$	28.6500	1.29826	0.649128	+	0.760680i	$0.275134\pi$
$488$	0	0
$489$	−39.8577	−1.80243
$490$	0	0
$491$	31.6220	1.42708	0.713541	−	0.700614i	$-0.247090\pi$
$491$	31.6220	1.42708	0.713541	+	0.700614i	$0.247090\pi$
$492$	0	0
$493$	17.3883	0.783129
$494$	0	0
$495$	−3.29277	−0.147999
$496$	0	0
$497$	25.5401	1.14563
$498$	0	0
$499$	17.0030	0.761160	0.380580	−	0.924748i	$-0.375724\pi$
$499$	17.0030	0.761160	0.380580	+	0.924748i	$0.375724\pi$
$500$	0	0
$501$	18.3747	0.820920
$502$	0	0
$503$	22.4988	1.00317	0.501587	−	0.865107i	$-0.332750\pi$
$503$	22.4988	1.00317	0.501587	+	0.865107i	$0.332750\pi$
$504$	0	0
$505$	−5.84890	−0.260273
$506$	0	0
$507$	4.45390	0.197805
$508$	0	0
$509$	3.67071	0.162701	0.0813507	−	0.996686i	$-0.474077\pi$
$509$	3.67071	0.162701	0.0813507	+	0.996686i	$0.474077\pi$
$510$	0	0
$511$	43.4168	1.92065
$512$	0	0
$513$	−10.6466	−0.470057
$514$	0	0
$515$	−0.802516	−0.0353631
$516$	0	0
$517$	−10.1719	−0.447360
$518$	0	0
$519$	−36.2966	−1.59325
$520$	0	0
$521$	−2.12912	−0.0932783	−0.0466391	−	0.998912i	$-0.514851\pi$
$521$	−2.12912	−0.0932783	−0.0466391	+	0.998912i	$0.514851\pi$
$522$	0	0
$523$	−17.2832	−0.755741	−0.377870	−	0.925859i	$-0.623343\pi$
$523$	−17.2832	−0.755741	−0.377870	+	0.925859i	$0.623343\pi$
$524$	0	0
$525$	−43.9655	−1.91881
$526$	0	0
$527$	27.9892	1.21923
$528$	0	0
$529$	−5.21729	−0.226839
$530$	0	0
$531$	6.73407	0.292234
$532$	0	0
$533$	−8.15324	−0.353156
$534$	0	0
$535$	4.39238	0.189899
$536$	0	0
$537$	−14.2478	−0.614840
$538$	0	0
$539$	−32.0161	−1.37903
$540$	0	0
$541$	26.6126	1.14416	0.572082	−	0.820196i	$-0.306136\pi$
$541$	26.6126	1.14416	0.572082	+	0.820196i	$0.306136\pi$
$542$	0	0
$543$	20.5281	0.880946
$544$	0	0
$545$	0.369221	0.0158157
$546$	0	0
$547$	−6.20409	−0.265268	−0.132634	−	0.991165i	$-0.542343\pi$
$547$	−6.20409	−0.265268	−0.132634	+	0.991165i	$0.542343\pi$
$548$	0	0
$549$	28.1383	1.20092
$550$	0	0
$551$	21.1552	0.901242
$552$	0	0
$553$	15.4454	0.656807
$554$	0	0
$555$	−4.08279	−0.173305
$556$	0	0
$557$	19.4296	0.823260	0.411630	−	0.911351i	$-0.364960\pi$
$557$	19.4296	0.823260	0.411630	+	0.911351i	$0.364960\pi$
$558$	0	0
$559$	−26.1316	−1.10525
$560$	0	0
$561$	−21.8131	−0.920950
$562$	0	0
$563$	−3.14872	−0.132703	−0.0663513	−	0.997796i	$-0.521136\pi$
$563$	−3.14872	−0.132703	−0.0663513	+	0.997796i	$0.521136\pi$
$564$	0	0
$565$	−4.72538	−0.198798
$566$	0	0
$567$	−47.8946	−2.01138
$568$	0	0
$569$	−6.93003	−0.290522	−0.145261	−	0.989393i	$-0.546402\pi$
$569$	−6.93003	−0.290522	−0.145261	+	0.989393i	$0.546402\pi$
$570$	0	0
$571$	14.8975	0.623441	0.311721	−	0.950174i	$-0.399095\pi$
$571$	14.8975	0.623441	0.311721	+	0.950174i	$0.399095\pi$
$572$	0	0
$573$	6.37703	0.266404
$574$	0	0
$575$	19.4043	0.809213
$576$	0	0
$577$	3.63261	0.151228	0.0756138	−	0.997137i	$-0.475908\pi$
$577$	3.63261	0.151228	0.0756138	+	0.997137i	$0.475908\pi$
$578$	0	0
$579$	−53.0519	−2.20476
$580$	0	0
$581$	−5.02627	−0.208525
$582$	0	0
$583$	8.54049	0.353711
$584$	0	0
$585$	−4.63962	−0.191825
$586$	0	0
$587$	−28.5083	−1.17666	−0.588331	−	0.808620i	$-0.700215\pi$
$587$	−28.5083	−1.17666	−0.588331	+	0.808620i	$0.700215\pi$
$588$	0	0
$589$	34.0527	1.40312
$590$	0	0
$591$	2.17045	0.0892804
$592$	0	0
$593$	3.85582	0.158340	0.0791698	−	0.996861i	$-0.474773\pi$
$593$	3.85582	0.158340	0.0791698	+	0.996861i	$0.474773\pi$
$594$	0	0
$595$	9.77080	0.400564
$596$	0	0
$597$	−40.6847	−1.66511
$598$	0	0
$599$	−31.1346	−1.27213	−0.636063	−	0.771637i	$-0.719438\pi$
$599$	−31.1346	−1.27213	−0.636063	+	0.771637i	$0.719438\pi$
$600$	0	0
$601$	−5.67089	−0.231320	−0.115660	−	0.993289i	$-0.536898\pi$
$601$	−5.67089	−0.231320	−0.115660	+	0.993289i	$0.536898\pi$
$602$	0	0
$603$	25.6596	1.04494
$604$	0	0
$605$	−2.17057	−0.0882462
$606$	0	0
$607$	37.6120	1.52662	0.763312	−	0.646030i	$-0.223572\pi$
$607$	37.6120	1.52662	0.763312	+	0.646030i	$0.223572\pi$
$608$	0	0
$609$	46.3467	1.87806
$610$	0	0
$611$	−14.3326	−0.579834
$612$	0	0
$613$	9.68281	0.391085	0.195542	−	0.980695i	$-0.437353\pi$
$613$	9.68281	0.391085	0.195542	+	0.980695i	$0.437353\pi$
$614$	0	0
$615$	2.93955	0.118534
$616$	0	0
$617$	−8.61903	−0.346989	−0.173495	−	0.984835i	$-0.555506\pi$
$617$	−8.61903	−0.346989	−0.173495	+	0.984835i	$0.555506\pi$
$618$	0	0
$619$	−32.4573	−1.30457	−0.652285	−	0.757973i	$-0.726190\pi$
$619$	−32.4573	−1.30457	−0.652285	+	0.757973i	$0.726190\pi$
$620$	0	0
$621$	10.2943	0.413095
$622$	0	0
$623$	5.02800	0.201443
$624$	0	0
$625$	19.1811	0.767243
$626$	0	0
$627$	−26.5386	−1.05985
$628$	0	0
$629$	−10.4768	−0.417737
$630$	0	0
$631$	48.5078	1.93107	0.965533	−	0.260282i	$-0.0838154\pi$
$631$	48.5078	1.93107	0.965533	+	0.260282i	$0.0838154\pi$
$632$	0	0
$633$	17.5135	0.696099
$634$	0	0
$635$	5.12435	0.203353
$636$	0	0
$637$	−45.1117	−1.78739
$638$	0	0
$639$	11.2201	0.443861
$640$	0	0
$641$	−26.4815	−1.04596	−0.522979	−	0.852346i	$-0.675179\pi$
$641$	−26.4815	−1.04596	−0.522979	+	0.852346i	$0.675179\pi$
$642$	0	0
$643$	−42.4550	−1.67426	−0.837130	−	0.547003i	$-0.815768\pi$
$643$	−42.4550	−1.67426	−0.837130	+	0.547003i	$0.815768\pi$
$644$	0	0
$645$	9.42142	0.370968
$646$	0	0
$647$	−32.1869	−1.26540	−0.632698	−	0.774399i	$-0.718053\pi$
$647$	−32.1869	−1.26540	−0.632698	+	0.774399i	$0.718053\pi$
$648$	0	0
$649$	−9.76237	−0.383207
$650$	0	0
$651$	74.6025	2.92390
$652$	0	0
$653$	27.2003	1.06443	0.532214	−	0.846610i	$-0.321360\pi$
$653$	27.2003	1.06443	0.532214	+	0.846610i	$0.321360\pi$
$654$	0	0
$655$	5.70957	0.223092
$656$	0	0
$657$	19.0736	0.744131
$658$	0	0
$659$	−40.7249	−1.58642	−0.793208	−	0.608951i	$-0.791591\pi$
$659$	−40.7249	−1.58642	−0.793208	+	0.608951i	$0.791591\pi$
$660$	0	0
$661$	−22.1828	−0.862811	−0.431406	−	0.902158i	$-0.641982\pi$
$661$	−22.1828	−0.862811	−0.431406	+	0.902158i	$0.641982\pi$
$662$	0	0
$663$	−30.7354	−1.19366
$664$	0	0
$665$	11.8875	0.460978
$666$	0	0
$667$	−20.4552	−0.792029
$668$	0	0
$669$	−24.4259	−0.944361
$670$	0	0
$671$	−40.7921	−1.57476
$672$	0	0
$673$	14.3194	0.551973	0.275986	−	0.961162i	$-0.410996\pi$
$673$	14.3194	0.551973	0.275986	+	0.961162i	$0.410996\pi$
$674$	0	0
$675$	11.2330	0.432357
$676$	0	0
$677$	18.9297	0.727528	0.363764	−	0.931491i	$-0.381491\pi$
$677$	18.9297	0.727528	0.363764	+	0.931491i	$0.381491\pi$
$678$	0	0
$679$	−54.1063	−2.07641
$680$	0	0
$681$	27.6575	1.05984
$682$	0	0
$683$	12.9336	0.494891	0.247445	−	0.968902i	$-0.420409\pi$
$683$	12.9336	0.494891	0.247445	+	0.968902i	$0.420409\pi$
$684$	0	0
$685$	5.76632	0.220320
$686$	0	0
$687$	−63.8892	−2.43752
$688$	0	0
$689$	12.0338	0.458453
$690$	0	0
$691$	−44.8246	−1.70521	−0.852605	−	0.522556i	$-0.824978\pi$
$691$	−44.8246	−1.70521	−0.852605	+	0.522556i	$0.824978\pi$
$692$	0	0
$693$	−22.5214	−0.855516
$694$	0	0
$695$	6.10983	0.231759
$696$	0	0
$697$	7.54314	0.285717
$698$	0	0
$699$	−51.1281	−1.93384
$700$	0	0
$701$	49.4281	1.86687	0.933436	−	0.358744i	$-0.116795\pi$
$701$	49.4281	1.86687	0.933436	+	0.358744i	$0.116795\pi$
$702$	0	0
$703$	−12.7464	−0.480741
$704$	0	0
$705$	5.16744	0.194617
$706$	0	0
$707$	−40.0044	−1.50452
$708$	0	0
$709$	13.0612	0.490523	0.245261	−	0.969457i	$-0.421126\pi$
$709$	13.0612	0.490523	0.245261	+	0.969457i	$0.421126\pi$
$710$	0	0
$711$	6.78538	0.254472
$712$	0	0
$713$	−32.9260	−1.23309
$714$	0	0
$715$	6.72605	0.251540
$716$	0	0
$717$	−35.5718	−1.32845
$718$	0	0
$719$	2.99554	0.111715	0.0558574	−	0.998439i	$-0.482211\pi$
$719$	2.99554	0.111715	0.0558574	+	0.998439i	$0.482211\pi$
$720$	0	0
$721$	−5.48893	−0.204418
$722$	0	0
$723$	7.08613	0.263536
$724$	0	0
$725$	−22.3204	−0.828961
$726$	0	0
$727$	1.96927	0.0730360	0.0365180	−	0.999333i	$-0.488373\pi$
$727$	1.96927	0.0730360	0.0365180	+	0.999333i	$0.488373\pi$
$728$	0	0
$729$	−4.83471	−0.179063
$730$	0	0
$731$	24.1762	0.894188
$732$	0	0
$733$	12.4034	0.458131	0.229065	−	0.973411i	$-0.426433\pi$
$733$	12.4034	0.458131	0.229065	+	0.973411i	$0.426433\pi$
$734$	0	0
$735$	16.2645	0.599924
$736$	0	0
$737$	−37.1987	−1.37023
$738$	0	0
$739$	17.1146	0.629570	0.314785	−	0.949163i	$-0.398068\pi$
$739$	17.1146	0.629570	0.314785	+	0.949163i	$0.398068\pi$
$740$	0	0
$741$	−37.3938	−1.37370
$742$	0	0
$743$	4.63712	0.170119	0.0850597	−	0.996376i	$-0.472892\pi$
$743$	4.63712	0.170119	0.0850597	+	0.996376i	$0.472892\pi$
$744$	0	0
$745$	2.45770	0.0900433
$746$	0	0
$747$	−2.20811	−0.0807904
$748$	0	0
$749$	30.0423	1.09772
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−51.2051	−1.86602
$754$	0	0
$755$	−5.57864	−0.203027
$756$	0	0
$757$	−36.3014	−1.31940	−0.659698	−	0.751531i	$-0.729316\pi$
$757$	−36.3014	−1.31940	−0.659698	+	0.751531i	$0.729316\pi$
$758$	0	0
$759$	25.6605	0.931417
$760$	0	0
$761$	31.0581	1.12586	0.562928	−	0.826506i	$-0.309675\pi$
$761$	31.0581	1.12586	0.562928	+	0.826506i	$0.309675\pi$
$762$	0	0
$763$	2.52534	0.0914235
$764$	0	0
$765$	4.29244	0.155194
$766$	0	0
$767$	−13.7555	−0.496683
$768$	0	0
$769$	25.1935	0.908501	0.454251	−	0.890874i	$-0.349907\pi$
$769$	25.1935	0.908501	0.454251	+	0.890874i	$0.349907\pi$
$770$	0	0
$771$	−7.69482	−0.277122
$772$	0	0
$773$	38.5118	1.38517	0.692586	−	0.721335i	$-0.256471\pi$
$773$	38.5118	1.38517	0.692586	+	0.721335i	$0.256471\pi$
$774$	0	0
$775$	−35.9284	−1.29058
$776$	0	0
$777$	−27.9249	−1.00180
$778$	0	0
$779$	9.17726	0.328809
$780$	0	0
$781$	−16.2658	−0.582035
$782$	0	0
$783$	−11.8414	−0.423176
$784$	0	0
$785$	7.25132	0.258811
$786$	0	0
$787$	−9.50028	−0.338649	−0.169324	−	0.985560i	$-0.554159\pi$
$787$	−9.50028	−0.338649	−0.169324	+	0.985560i	$0.554159\pi$
$788$	0	0
$789$	−36.0031	−1.28174
$790$	0	0
$791$	−32.3200	−1.14917
$792$	0	0
$793$	−57.4775	−2.04109
$794$	0	0
$795$	−4.33866	−0.153876
$796$	0	0
$797$	23.4926	0.832152	0.416076	−	0.909330i	$-0.363405\pi$
$797$	23.4926	0.832152	0.416076	+	0.909330i	$0.363405\pi$
$798$	0	0
$799$	13.2601	0.469108
$800$	0	0
$801$	2.20887	0.0780465
$802$	0	0
$803$	−27.6509	−0.975781
$804$	0	0
$805$	−11.4942	−0.405116
$806$	0	0
$807$	−9.29958	−0.327361
$808$	0	0
$809$	−0.398415	−0.0140075	−0.00700376	−	0.999975i	$-0.502229\pi$
$809$	−0.398415	−0.0140075	−0.00700376	+	0.999975i	$0.502229\pi$
$810$	0	0
$811$	2.79876	0.0982778	0.0491389	−	0.998792i	$-0.484352\pi$
$811$	2.79876	0.0982778	0.0491389	+	0.998792i	$0.484352\pi$
$812$	0	0
$813$	−22.5024	−0.789195
$814$	0	0
$815$	−11.3704	−0.398288
$816$	0	0
$817$	29.4136	1.02905
$818$	0	0
$819$	−31.7334	−1.10885
$820$	0	0
$821$	−38.0781	−1.32894	−0.664468	−	0.747317i	$-0.731342\pi$
$821$	−38.0781	−1.32894	−0.664468	+	0.747317i	$0.731342\pi$
$822$	0	0
$823$	4.59598	0.160206	0.0801029	−	0.996787i	$-0.474475\pi$
$823$	4.59598	0.160206	0.0801029	+	0.996787i	$0.474475\pi$
$824$	0	0
$825$	28.0004	0.974848
$826$	0	0
$827$	−3.42383	−0.119058	−0.0595291	−	0.998227i	$-0.518960\pi$
$827$	−3.42383	−0.119058	−0.0595291	+	0.998227i	$0.518960\pi$
$828$	0	0
$829$	−5.34995	−0.185811	−0.0929057	−	0.995675i	$-0.529616\pi$
$829$	−5.34995	−0.185811	−0.0929057	+	0.995675i	$0.529616\pi$
$830$	0	0
$831$	−0.360332	−0.0124998
$832$	0	0
$833$	41.7360	1.44607
$834$	0	0
$835$	5.24185	0.181402
$836$	0	0
$837$	−19.0606	−0.658830
$838$	0	0
$839$	−36.7103	−1.26738	−0.633690	−	0.773587i	$-0.718461\pi$
$839$	−36.7103	−1.26738	−0.633690	+	0.773587i	$0.718461\pi$
$840$	0	0
$841$	−5.47065	−0.188643
$842$	0	0
$843$	−21.3492	−0.735307
$844$	0	0
$845$	1.27059	0.0437096
$846$	0	0
$847$	−14.8459	−0.510112
$848$	0	0
$849$	30.8184	1.05769
$850$	0	0
$851$	12.3247	0.422485
$852$	0	0
$853$	−7.10322	−0.243210	−0.121605	−	0.992579i	$-0.538804\pi$
$853$	−7.10322	−0.243210	−0.121605	+	0.992579i	$0.538804\pi$
$854$	0	0
$855$	5.22234	0.178600
$856$	0	0
$857$	27.0754	0.924879	0.462439	−	0.886651i	$-0.346974\pi$
$857$	27.0754	0.924879	0.462439	+	0.886651i	$0.346974\pi$
$858$	0	0
$859$	−26.9035	−0.917935	−0.458967	−	0.888453i	$-0.651780\pi$
$859$	−26.9035	−0.917935	−0.458967	+	0.888453i	$0.651780\pi$
$860$	0	0
$861$	20.1055	0.685194
$862$	0	0
$863$	−13.7747	−0.468895	−0.234447	−	0.972129i	$-0.575328\pi$
$863$	−13.7747	−0.468895	−0.234447	+	0.972129i	$0.575328\pi$
$864$	0	0
$865$	−10.3545	−0.352065
$866$	0	0
$867$	−9.18345	−0.311886
$868$	0	0
$869$	−9.83676	−0.333689
$870$	0	0
$871$	−52.4143	−1.77599
$872$	0	0
$873$	−23.7696	−0.804479
$874$	0	0
$875$	−26.1708	−0.884735
$876$	0	0
$877$	−16.4117	−0.554182	−0.277091	−	0.960844i	$-0.589370\pi$
$877$	−16.4117	−0.554182	−0.277091	+	0.960844i	$0.589370\pi$
$878$	0	0
$879$	−13.5483	−0.456971
$880$	0	0
$881$	39.4153	1.32794	0.663968	−	0.747761i	$-0.268871\pi$
$881$	39.4153	1.32794	0.663968	+	0.747761i	$0.268871\pi$
$882$	0	0
$883$	7.22326	0.243082	0.121541	−	0.992586i	$-0.461216\pi$
$883$	7.22326	0.243082	0.121541	+	0.992586i	$0.461216\pi$
$884$	0	0
$885$	4.95938	0.166708
$886$	0	0
$887$	−2.90973	−0.0976993	−0.0488497	−	0.998806i	$-0.515556\pi$
$887$	−2.90973	−0.0976993	−0.0488497	+	0.998806i	$0.515556\pi$
$888$	0	0
$889$	35.0487	1.17550
$890$	0	0
$891$	30.5027	1.02188
$892$	0	0
$893$	16.1327	0.539860
$894$	0	0
$895$	−4.06456	−0.135863
$896$	0	0
$897$	36.1565	1.20723
$898$	0	0
$899$	37.8743	1.26318
$900$	0	0
$901$	−11.1334	−0.370906
$902$	0	0
$903$	64.4392	2.14440
$904$	0	0
$905$	5.85617	0.194666
$906$	0	0
$907$	17.8466	0.592588	0.296294	−	0.955097i	$-0.404249\pi$
$907$	17.8466	0.592588	0.296294	+	0.955097i	$0.404249\pi$
$908$	0	0
$909$	−17.5745	−0.582909
$910$	0	0
$911$	33.7075	1.11678	0.558390	−	0.829578i	$-0.311419\pi$
$911$	33.7075	1.11678	0.558390	+	0.829578i	$0.311419\pi$
$912$	0	0
$913$	3.20109	0.105941
$914$	0	0
$915$	20.7228	0.685075
$916$	0	0
$917$	39.0515	1.28959
$918$	0	0
$919$	29.6883	0.979328	0.489664	−	0.871911i	$-0.337119\pi$
$919$	29.6883	0.979328	0.489664	+	0.871911i	$0.337119\pi$
$920$	0	0
$921$	41.7669	1.37627
$922$	0	0
$923$	−22.9190	−0.754389
$924$	0	0
$925$	13.4485	0.442185
$926$	0	0
$927$	−2.41136	−0.0791994
$928$	0	0
$929$	43.0337	1.41189	0.705945	−	0.708266i	$-0.250522\pi$
$929$	43.0337	1.41189	0.705945	+	0.708266i	$0.250522\pi$
$930$	0	0
$931$	50.7776	1.66417
$932$	0	0
$933$	−29.5797	−0.968395
$934$	0	0
$935$	−6.22275	−0.203506
$936$	0	0
$937$	52.8530	1.72663	0.863317	−	0.504663i	$-0.168383\pi$
$937$	52.8530	1.72663	0.863317	+	0.504663i	$0.168383\pi$
$938$	0	0
$939$	15.9943	0.521955
$940$	0	0
$941$	44.6043	1.45406	0.727029	−	0.686607i	$-0.240901\pi$
$941$	44.6043	1.45406	0.727029	+	0.686607i	$0.240901\pi$
$942$	0	0
$943$	−8.87360	−0.288964
$944$	0	0
$945$	−6.65389	−0.216451
$946$	0	0
$947$	−34.6742	−1.12676	−0.563380	−	0.826198i	$-0.690499\pi$
$947$	−34.6742	−1.12676	−0.563380	+	0.826198i	$0.690499\pi$
$948$	0	0
$949$	−38.9611	−1.26473
$950$	0	0
$951$	27.5085	0.892024
$952$	0	0
$953$	−22.9288	−0.742735	−0.371368	−	0.928486i	$-0.621111\pi$
$953$	−22.9288	−0.742735	−0.371368	+	0.928486i	$0.621111\pi$
$954$	0	0
$955$	1.81921	0.0588683
$956$	0	0
$957$	−29.5169	−0.954146
$958$	0	0
$959$	39.4396	1.27357
$960$	0	0
$961$	29.9648	0.966605
$962$	0	0
$963$	13.1980	0.425299
$964$	0	0
$965$	−15.1344	−0.487194
$966$	0	0
$967$	29.9139	0.961965	0.480982	−	0.876730i	$-0.340280\pi$
$967$	29.9139	0.961965	0.480982	+	0.876730i	$0.340280\pi$
$968$	0	0
$969$	34.5957	1.11137
$970$	0	0
$971$	−14.3020	−0.458973	−0.229487	−	0.973312i	$-0.573705\pi$
$971$	−14.3020	−0.458973	−0.229487	+	0.973312i	$0.573705\pi$
$972$	0	0
$973$	41.7891	1.33970
$974$	0	0
$975$	39.4535	1.26352
$976$	0	0
$977$	−29.3000	−0.937391	−0.468696	−	0.883360i	$-0.655276\pi$
$977$	−29.3000	−0.937391	−0.468696	+	0.883360i	$0.655276\pi$
$978$	0	0
$979$	−3.20219	−0.102342
$980$	0	0
$981$	1.10941	0.0354209
$982$	0	0
$983$	−12.9742	−0.413813	−0.206907	−	0.978361i	$-0.566340\pi$
$983$	−12.9742	−0.413813	−0.206907	+	0.978361i	$0.566340\pi$
$984$	0	0
$985$	0.619177	0.0197286
$986$	0	0
$987$	35.3434	1.12499
$988$	0	0
$989$	−28.4403	−0.904350
$990$	0	0
$991$	43.0365	1.36710	0.683549	−	0.729905i	$-0.260436\pi$
$991$	43.0365	1.36710	0.683549	+	0.729905i	$0.260436\pi$
$992$	0	0
$993$	−44.3387	−1.40705
$994$	0	0
$995$	−11.6063	−0.367946
$996$	0	0
$997$	−51.1614	−1.62030	−0.810149	−	0.586224i	$-0.800614\pi$
$997$	−51.1614	−1.62030	−0.810149	+	0.586224i	$0.800614\pi$
$998$	0	0
$999$	7.13467	0.225731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.41	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.41	✓	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)

\(f(q)\)	\(=\)	\(q+2.21288 q^{3} +0.631281 q^{5} +4.31774 q^{7} +1.89684 q^{9} +O(q^{10})\)	\(q+2.21288 q^{3} +0.631281 q^{5} +4.31774 q^{7} +1.89684 q^{9} -2.74984 q^{11} -3.87462 q^{13} +1.39695 q^{15} +3.58469 q^{17} +4.36126 q^{19} +9.55464 q^{21} -4.21696 q^{23} -4.60148 q^{25} -2.44116 q^{27} +4.85071 q^{29} +7.80799 q^{31} -6.08508 q^{33} +2.72570 q^{35} -2.92265 q^{37} -8.57408 q^{39} +2.10427 q^{41} +6.74428 q^{43} +1.19744 q^{45} +3.69909 q^{47} +11.6429 q^{49} +7.93249 q^{51} -3.10581 q^{53} -1.73592 q^{55} +9.65095 q^{57} +3.55015 q^{59} +14.8343 q^{61} +8.19005 q^{63} -2.44598 q^{65} +13.5276 q^{67} -9.33162 q^{69} +5.91516 q^{71} +10.0555 q^{73} -10.1825 q^{75} -11.8731 q^{77} +3.57721 q^{79} -11.0925 q^{81} -1.16410 q^{83} +2.26295 q^{85} +10.7340 q^{87} +1.16450 q^{89} -16.7296 q^{91} +17.2782 q^{93} +2.75318 q^{95} -12.5312 q^{97} -5.21601 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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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