LMFDB - Embedded newform 312.4.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [312,4,Mod(1,312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("312.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,-4,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.4085959218$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +3.29150 q^{5} -25.8745 q^{7} +9.00000 q^{9} -8.83399 q^{11} -13.0000 q^{13} +9.87451 q^{15} -10.2510 q^{17} -119.956 q^{19} -77.6235 q^{21} -141.830 q^{23} -114.166 q^{25} +27.0000 q^{27} +170.826 q^{29} +226.450 q^{31} -26.5020 q^{33} -85.1660 q^{35} -225.409 q^{37} -39.0000 q^{39} -274.701 q^{41} -111.247 q^{43} +29.6235 q^{45} +156.405 q^{47} +326.490 q^{49} -30.7530 q^{51} -85.1581 q^{53} -29.0771 q^{55} -359.867 q^{57} -889.911 q^{59} +463.668 q^{61} -232.871 q^{63} -42.7895 q^{65} -459.041 q^{67} -425.490 q^{69} -560.996 q^{71} +784.891 q^{73} -342.498 q^{75} +228.575 q^{77} +241.830 q^{79} +81.0000 q^{81} -1271.56 q^{83} -33.7411 q^{85} +512.478 q^{87} +1085.02 q^{89} +336.369 q^{91} +679.349 q^{93} -394.834 q^{95} -79.9032 q^{97} -79.5059 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 4 q^{5} - 20 q^{7} + 18 q^{9} - 60 q^{11} - 26 q^{13} - 12 q^{15} - 84 q^{17} - 60 q^{19} - 60 q^{21} - 72 q^{23} - 186 q^{25} + 54 q^{27} - 124 q^{29} - 108 q^{31} - 180 q^{33} - 128 q^{35}+ \cdots - 540 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 4 * q^5 - 20 * q^7 + 18 * q^9 - 60 * q^11 - 26 * q^13 - 12 * q^15 - 84 * q^17 - 60 * q^19 - 60 * q^21 - 72 * q^23 - 186 * q^25 + 54 * q^27 - 124 * q^29 - 108 * q^31 - 180 * q^33 - 128 * q^35 + 36 * q^37 - 78 * q^39 - 52 * q^41 - 32 * q^43 - 36 * q^45 - 428 * q^47 + 18 * q^49 - 252 * q^51 + 380 * q^53 + 344 * q^55 - 180 * q^57 - 1420 * q^59 + 1012 * q^61 - 180 * q^63 + 52 * q^65 - 844 * q^67 - 216 * q^69 - 868 * q^71 - 60 * q^73 - 558 * q^75 - 72 * q^77 + 272 * q^79 + 162 * q^81 - 1252 * q^83 + 504 * q^85 - 372 * q^87 + 572 * q^89 + 260 * q^91 - 324 * q^93 - 832 * q^95 + 708 * q^97 - 540 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	3.29150	0.294401	0.147200	−	0.989107i	$-0.452974\pi$
$5$	3.29150	0.294401	0.147200	+	0.989107i	$0.452974\pi$
$6$	0	0
$7$	−25.8745	−1.39709	−0.698546	−	0.715565i	$-0.746169\pi$
$7$	−25.8745	−1.39709	−0.698546	+	0.715565i	$0.746169\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−8.83399	−0.242141	−0.121070	−	0.992644i	$-0.538633\pi$
$11$	−8.83399	−0.242141	−0.121070	+	0.992644i	$0.538633\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	9.87451	0.169972
$16$	0	0
$17$	−10.2510	−0.146249	−0.0731244	−	0.997323i	$-0.523297\pi$
$17$	−10.2510	−0.146249	−0.0731244	+	0.997323i	$0.523297\pi$
$18$	0	0
$19$	−119.956	−1.44840	−0.724202	−	0.689588i	$-0.757792\pi$
$19$	−119.956	−1.44840	−0.724202	+	0.689588i	$0.757792\pi$
$20$	0	0
$21$	−77.6235	−0.806612
$22$	0	0
$23$	−141.830	−1.28581	−0.642905	−	0.765946i	$-0.722271\pi$
$23$	−141.830	−1.28581	−0.642905	+	0.765946i	$0.722271\pi$
$24$	0	0
$25$	−114.166	−0.913328
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	170.826	1.09385	0.546924	−	0.837182i	$-0.315799\pi$
$29$	170.826	1.09385	0.546924	+	0.837182i	$0.315799\pi$
$30$	0	0
$31$	226.450	1.31199	0.655993	−	0.754767i	$-0.272250\pi$
$31$	226.450	1.31199	0.655993	+	0.754767i	$0.272250\pi$
$32$	0	0
$33$	−26.5020	−0.139800
$34$	0	0
$35$	−85.1660	−0.411305
$36$	0	0
$37$	−225.409	−1.00154	−0.500771	−	0.865580i	$-0.666950\pi$
$37$	−225.409	−1.00154	−0.500771	+	0.865580i	$0.666950\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−274.701	−1.04637	−0.523183	−	0.852220i	$-0.675256\pi$
$41$	−274.701	−1.04637	−0.523183	+	0.852220i	$0.675256\pi$
$42$	0	0
$43$	−111.247	−0.394535	−0.197268	−	0.980350i	$-0.563207\pi$
$43$	−111.247	−0.394535	−0.197268	+	0.980350i	$0.563207\pi$
$44$	0	0
$45$	29.6235	0.0981336
$46$	0	0
$47$	156.405	0.485405	0.242703	−	0.970101i	$-0.421966\pi$
$47$	156.405	0.485405	0.242703	+	0.970101i	$0.421966\pi$
$48$	0	0
$49$	326.490	0.951866
$50$	0	0
$51$	−30.7530	−0.0844368
$52$	0	0
$53$	−85.1581	−0.220705	−0.110353	−	0.993893i	$-0.535198\pi$
$53$	−85.1581	−0.220705	−0.110353	+	0.993893i	$0.535198\pi$
$54$	0	0
$55$	−29.0771	−0.0712865
$56$	0	0
$57$	−359.867	−0.836237
$58$	0	0
$59$	−889.911	−1.96367	−0.981835	−	0.189736i	$-0.939237\pi$
$59$	−889.911	−1.96367	−0.981835	+	0.189736i	$0.939237\pi$
$60$	0	0
$61$	463.668	0.973223	0.486611	−	0.873618i	$-0.338233\pi$
$61$	463.668	0.973223	0.486611	+	0.873618i	$0.338233\pi$
$62$	0	0
$63$	−232.871	−0.465697
$64$	0	0
$65$	−42.7895	−0.0816521
$66$	0	0
$67$	−459.041	−0.837026	−0.418513	−	0.908211i	$-0.637449\pi$
$67$	−459.041	−0.837026	−0.418513	+	0.908211i	$0.637449\pi$
$68$	0	0
$69$	−425.490	−0.742362
$70$	0	0
$71$	−560.996	−0.937718	−0.468859	−	0.883273i	$-0.655335\pi$
$71$	−560.996	−0.937718	−0.468859	+	0.883273i	$0.655335\pi$
$72$	0	0
$73$	784.891	1.25842	0.629210	−	0.777236i	$-0.283379\pi$
$73$	784.891	1.25842	0.629210	+	0.777236i	$0.283379\pi$
$74$	0	0
$75$	−342.498	−0.527310
$76$	0	0
$77$	228.575	0.338293
$78$	0	0
$79$	241.830	0.344405	0.172203	−	0.985062i	$-0.444912\pi$
$79$	241.830	0.344405	0.172203	+	0.985062i	$0.444912\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1271.56	−1.68159	−0.840796	−	0.541351i	$-0.817913\pi$
$83$	−1271.56	−1.68159	−0.840796	+	0.541351i	$0.817913\pi$
$84$	0	0
$85$	−33.7411	−0.0430558
$86$	0	0
$87$	512.478	0.631534
$88$	0	0
$89$	1085.02	1.29226	0.646132	−	0.763225i	$-0.276385\pi$
$89$	1085.02	1.29226	0.646132	+	0.763225i	$0.276385\pi$
$90$	0	0
$91$	336.369	0.387484
$92$	0	0
$93$	679.349	0.757475
$94$	0	0
$95$	−394.834	−0.426412
$96$	0	0
$97$	−79.9032	−0.0836386	−0.0418193	−	0.999125i	$-0.513315\pi$
$97$	−79.9032	−0.0836386	−0.0418193	+	0.999125i	$0.513315\pi$
$98$	0	0
$99$	−79.5059	−0.0807136
$100$	0	0
$101$	−195.725	−0.192826	−0.0964129	−	0.995341i	$-0.530737\pi$
$101$	−195.725	−0.192826	−0.0964129	+	0.995341i	$0.530737\pi$
$102$	0	0
$103$	899.425	0.860417	0.430209	−	0.902730i	$-0.358440\pi$
$103$	899.425	0.860417	0.430209	+	0.902730i	$0.358440\pi$
$104$	0	0
$105$	−255.498	−0.237467
$106$	0	0
$107$	499.304	0.451118	0.225559	−	0.974230i	$-0.427579\pi$
$107$	499.304	0.451118	0.225559	+	0.974230i	$0.427579\pi$
$108$	0	0
$109$	1952.82	1.71602	0.858009	−	0.513634i	$-0.171701\pi$
$109$	1952.82	1.71602	0.858009	+	0.513634i	$0.171701\pi$
$110$	0	0
$111$	−676.227	−0.578240
$112$	0	0
$113$	1284.17	1.06907	0.534533	−	0.845147i	$-0.320487\pi$
$113$	1284.17	1.06907	0.534533	+	0.845147i	$0.320487\pi$
$114$	0	0
$115$	−466.834	−0.378543
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	265.239	0.204323
$120$	0	0
$121$	−1252.96	−0.941368
$122$	0	0
$123$	−824.102	−0.604120
$124$	0	0
$125$	−787.216	−0.563286
$126$	0	0
$127$	−1397.69	−0.976577	−0.488288	−	0.872682i	$-0.662379\pi$
$127$	−1397.69	−0.976577	−0.488288	+	0.872682i	$0.662379\pi$
$128$	0	0
$129$	−333.741	−0.227785
$130$	0	0
$131$	942.138	0.628359	0.314179	−	0.949364i	$-0.398271\pi$
$131$	942.138	0.628359	0.314179	+	0.949364i	$0.398271\pi$
$132$	0	0
$133$	3103.79	2.02355
$134$	0	0
$135$	88.8706	0.0566575
$136$	0	0
$137$	−827.380	−0.515970	−0.257985	−	0.966149i	$-0.583059\pi$
$137$	−827.380	−0.515970	−0.257985	+	0.966149i	$0.583059\pi$
$138$	0	0
$139$	1248.49	0.761836	0.380918	−	0.924609i	$-0.375608\pi$
$139$	1248.49	0.761836	0.380918	+	0.924609i	$0.375608\pi$
$140$	0	0
$141$	469.216	0.280249
$142$	0	0
$143$	114.842	0.0671578
$144$	0	0
$145$	562.275	0.322030
$146$	0	0
$147$	979.470	0.549560
$148$	0	0
$149$	−222.936	−0.122575	−0.0612873	−	0.998120i	$-0.519521\pi$
$149$	−222.936	−0.122575	−0.0612873	+	0.998120i	$0.519521\pi$
$150$	0	0
$151$	2077.67	1.11972	0.559861	−	0.828587i	$-0.310855\pi$
$151$	2077.67	1.11972	0.559861	+	0.828587i	$0.310855\pi$
$152$	0	0
$153$	−92.2589	−0.0487496
$154$	0	0
$155$	745.360	0.386250
$156$	0	0
$157$	276.201	0.140403	0.0702015	−	0.997533i	$-0.477636\pi$
$157$	276.201	0.140403	0.0702015	+	0.997533i	$0.477636\pi$
$158$	0	0
$159$	−255.474	−0.127424
$160$	0	0
$161$	3669.78	1.79639
$162$	0	0
$163$	3094.60	1.48704	0.743520	−	0.668714i	$-0.233155\pi$
$163$	3094.60	1.48704	0.743520	+	0.668714i	$0.233155\pi$
$164$	0	0
$165$	−87.2313	−0.0411573
$166$	0	0
$167$	−3239.52	−1.50109	−0.750543	−	0.660822i	$-0.770208\pi$
$167$	−3239.52	−1.50109	−0.750543	+	0.660822i	$0.770208\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1079.60	−0.482802
$172$	0	0
$173$	−314.607	−0.138261	−0.0691303	−	0.997608i	$-0.522022\pi$
$173$	−314.607	−0.138261	−0.0691303	+	0.997608i	$0.522022\pi$
$174$	0	0
$175$	2953.99	1.27600
$176$	0	0
$177$	−2669.73	−1.13373
$178$	0	0
$179$	2745.22	1.14630	0.573149	−	0.819451i	$-0.305722\pi$
$179$	2745.22	1.14630	0.573149	+	0.819451i	$0.305722\pi$
$180$	0	0
$181$	3261.14	1.33922	0.669610	−	0.742713i	$-0.266461\pi$
$181$	3261.14	1.33922	0.669610	+	0.742713i	$0.266461\pi$
$182$	0	0
$183$	1391.00	0.561891
$184$	0	0
$185$	−741.935	−0.294855
$186$	0	0
$187$	90.5571	0.0354128
$188$	0	0
$189$	−698.612	−0.268871
$190$	0	0
$191$	−2432.32	−0.921449	−0.460725	−	0.887543i	$-0.652410\pi$
$191$	−2432.32	−0.921449	−0.460725	+	0.887543i	$0.652410\pi$
$192$	0	0
$193$	−971.846	−0.362461	−0.181231	−	0.983441i	$-0.558008\pi$
$193$	−971.846	−0.362461	−0.181231	+	0.983441i	$0.558008\pi$
$194$	0	0
$195$	−128.369	−0.0471419
$196$	0	0
$197$	1881.11	0.680321	0.340161	−	0.940367i	$-0.389519\pi$
$197$	1881.11	0.680321	0.340161	+	0.940367i	$0.389519\pi$
$198$	0	0
$199$	1558.88	0.555305	0.277653	−	0.960682i	$-0.410444\pi$
$199$	1558.88	0.555305	0.277653	+	0.960682i	$0.410444\pi$
$200$	0	0
$201$	−1377.12	−0.483257
$202$	0	0
$203$	−4420.04	−1.52821
$204$	0	0
$205$	−904.178	−0.308051
$206$	0	0
$207$	−1276.47	−0.428603
$208$	0	0
$209$	1059.69	0.350718
$210$	0	0
$211$	−2227.58	−0.726792	−0.363396	−	0.931635i	$-0.618383\pi$
$211$	−2227.58	−0.726792	−0.363396	+	0.931635i	$0.618383\pi$
$212$	0	0
$213$	−1682.99	−0.541392
$214$	0	0
$215$	−366.170	−0.116152
$216$	0	0
$217$	−5859.27	−1.83296
$218$	0	0
$219$	2354.67	0.726549
$220$	0	0
$221$	133.263	0.0405621
$222$	0	0
$223$	4031.20	1.21053	0.605267	−	0.796023i	$-0.293066\pi$
$223$	4031.20	1.21053	0.605267	+	0.796023i	$0.293066\pi$
$224$	0	0
$225$	−1027.49	−0.304443
$226$	0	0
$227$	−5877.53	−1.71853	−0.859264	−	0.511533i	$-0.829078\pi$
$227$	−5877.53	−1.71853	−0.859264	+	0.511533i	$0.829078\pi$
$228$	0	0
$229$	−3802.10	−1.09716	−0.548580	−	0.836098i	$-0.684832\pi$
$229$	−3802.10	−1.09716	−0.548580	+	0.836098i	$0.684832\pi$
$230$	0	0
$231$	685.725	0.195313
$232$	0	0
$233$	−3552.99	−0.998988	−0.499494	−	0.866317i	$-0.666481\pi$
$233$	−3552.99	−0.998988	−0.499494	+	0.866317i	$0.666481\pi$
$234$	0	0
$235$	514.808	0.142904
$236$	0	0
$237$	725.490	0.198842
$238$	0	0
$239$	−1307.33	−0.353824	−0.176912	−	0.984227i	$-0.556611\pi$
$239$	−1307.33	−0.353824	−0.176912	+	0.984227i	$0.556611\pi$
$240$	0	0
$241$	−2787.48	−0.745051	−0.372526	−	0.928022i	$-0.621508\pi$
$241$	−2787.48	−0.745051	−0.372526	+	0.928022i	$0.621508\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1074.64	0.280230
$246$	0	0
$247$	1559.42	0.401715
$248$	0	0
$249$	−3814.69	−0.970868
$250$	0	0
$251$	−727.383	−0.182916	−0.0914582	−	0.995809i	$-0.529153\pi$
$251$	−727.383	−0.182916	−0.0914582	+	0.995809i	$0.529153\pi$
$252$	0	0
$253$	1252.93	0.311347
$254$	0	0
$255$	−101.223	−0.0248583
$256$	0	0
$257$	−2055.13	−0.498815	−0.249407	−	0.968399i	$-0.580236\pi$
$257$	−2055.13	−0.498815	−0.249407	+	0.968399i	$0.580236\pi$
$258$	0	0
$259$	5832.35	1.39925
$260$	0	0
$261$	1537.44	0.364616
$262$	0	0
$263$	−851.320	−0.199599	−0.0997997	−	0.995008i	$-0.531820\pi$
$263$	−851.320	−0.199599	−0.0997997	+	0.995008i	$0.531820\pi$
$264$	0	0
$265$	−280.298	−0.0649758
$266$	0	0
$267$	3255.05	0.746089
$268$	0	0
$269$	5598.13	1.26886	0.634431	−	0.772979i	$-0.281234\pi$
$269$	5598.13	1.26886	0.634431	+	0.772979i	$0.281234\pi$
$270$	0	0
$271$	−6243.82	−1.39958	−0.699788	−	0.714351i	$-0.746722\pi$
$271$	−6243.82	−1.39958	−0.699788	+	0.714351i	$0.746722\pi$
$272$	0	0
$273$	1009.11	0.223714
$274$	0	0
$275$	1008.54	0.221154
$276$	0	0
$277$	−5454.37	−1.18311	−0.591555	−	0.806265i	$-0.701486\pi$
$277$	−5454.37	−1.18311	−0.591555	+	0.806265i	$0.701486\pi$
$278$	0	0
$279$	2038.05	0.437329
$280$	0	0
$281$	−4747.09	−1.00778	−0.503892	−	0.863766i	$-0.668099\pi$
$281$	−4747.09	−1.00778	−0.503892	+	0.863766i	$0.668099\pi$
$282$	0	0
$283$	−1750.54	−0.367698	−0.183849	−	0.982954i	$-0.558856\pi$
$283$	−1750.54	−0.367698	−0.183849	+	0.982954i	$0.558856\pi$
$284$	0	0
$285$	−1184.50	−0.246189
$286$	0	0
$287$	7107.74	1.46187
$288$	0	0
$289$	−4807.92	−0.978611
$290$	0	0
$291$	−239.710	−0.0482888
$292$	0	0
$293$	2850.77	0.568408	0.284204	−	0.958764i	$-0.408271\pi$
$293$	2850.77	0.568408	0.284204	+	0.958764i	$0.408271\pi$
$294$	0	0
$295$	−2929.14	−0.578106
$296$	0	0
$297$	−238.518	−0.0466000
$298$	0	0
$299$	1843.79	0.356619
$300$	0	0
$301$	2878.46	0.551202
$302$	0	0
$303$	−587.176	−0.111328
$304$	0	0
$305$	1526.16	0.286518
$306$	0	0
$307$	−5557.02	−1.03308	−0.516541	−	0.856263i	$-0.672781\pi$
$307$	−5557.02	−1.03308	−0.516541	+	0.856263i	$0.672781\pi$
$308$	0	0
$309$	2698.27	0.496762
$310$	0	0
$311$	−4858.88	−0.885922	−0.442961	−	0.896541i	$-0.646072\pi$
$311$	−4858.88	−0.885922	−0.442961	+	0.896541i	$0.646072\pi$
$312$	0	0
$313$	−9615.77	−1.73647	−0.868236	−	0.496152i	$-0.834746\pi$
$313$	−9615.77	−1.73647	−0.868236	+	0.496152i	$0.834746\pi$
$314$	0	0
$315$	−766.494	−0.137102
$316$	0	0
$317$	782.094	0.138570	0.0692851	−	0.997597i	$-0.477928\pi$
$317$	782.094	0.138570	0.0692851	+	0.997597i	$0.477928\pi$
$318$	0	0
$319$	−1509.08	−0.264865
$320$	0	0
$321$	1497.91	0.260453
$322$	0	0
$323$	1229.66	0.211827
$324$	0	0
$325$	1484.16	0.253312
$326$	0	0
$327$	5858.45	0.990744
$328$	0	0
$329$	−4046.91	−0.678156
$330$	0	0
$331$	−1707.67	−0.283571	−0.141785	−	0.989897i	$-0.545284\pi$
$331$	−1707.67	−0.283571	−0.141785	+	0.989897i	$0.545284\pi$
$332$	0	0
$333$	−2028.68	−0.333847
$334$	0	0
$335$	−1510.93	−0.246421
$336$	0	0
$337$	5652.60	0.913699	0.456850	−	0.889544i	$-0.348978\pi$
$337$	5652.60	0.913699	0.456850	+	0.889544i	$0.348978\pi$
$338$	0	0
$339$	3852.51	0.617226
$340$	0	0
$341$	−2000.45	−0.317685
$342$	0	0
$343$	427.184	0.0672471
$344$	0	0
$345$	−1400.50	−0.218552
$346$	0	0
$347$	−235.759	−0.0364732	−0.0182366	−	0.999834i	$-0.505805\pi$
$347$	−235.759	−0.0364732	−0.0182366	+	0.999834i	$0.505805\pi$
$348$	0	0
$349$	−6805.12	−1.04375	−0.521876	−	0.853021i	$-0.674768\pi$
$349$	−6805.12	−1.04375	−0.521876	+	0.853021i	$0.674768\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	4558.89	0.687380	0.343690	−	0.939083i	$-0.388323\pi$
$353$	4558.89	0.687380	0.343690	+	0.939083i	$0.388323\pi$
$354$	0	0
$355$	−1846.52	−0.276065
$356$	0	0
$357$	795.718	0.117966
$358$	0	0
$359$	−10738.4	−1.57870	−0.789349	−	0.613944i	$-0.789582\pi$
$359$	−10738.4	−1.57870	−0.789349	+	0.613944i	$0.789582\pi$
$360$	0	0
$361$	7530.33	1.09788
$362$	0	0
$363$	−3758.88	−0.543499
$364$	0	0
$365$	2583.47	0.370480
$366$	0	0
$367$	11442.0	1.62743	0.813717	−	0.581261i	$-0.197440\pi$
$367$	11442.0	1.62743	0.813717	+	0.581261i	$0.197440\pi$
$368$	0	0
$369$	−2472.31	−0.348789
$370$	0	0
$371$	2203.42	0.308345
$372$	0	0
$373$	−1374.29	−0.190773	−0.0953863	−	0.995440i	$-0.530409\pi$
$373$	−1374.29	−0.190773	−0.0953863	+	0.995440i	$0.530409\pi$
$374$	0	0
$375$	−2361.65	−0.325213
$376$	0	0
$377$	−2220.74	−0.303379
$378$	0	0
$379$	3572.95	0.484248	0.242124	−	0.970245i	$-0.422156\pi$
$379$	3572.95	0.484248	0.242124	+	0.970245i	$0.422156\pi$
$380$	0	0
$381$	−4193.08	−0.563827
$382$	0	0
$383$	12109.5	1.61557	0.807787	−	0.589475i	$-0.200665\pi$
$383$	12109.5	1.61557	0.807787	+	0.589475i	$0.200665\pi$
$384$	0	0
$385$	752.356	0.0995938
$386$	0	0
$387$	−1001.22	−0.131512
$388$	0	0
$389$	3096.24	0.403562	0.201781	−	0.979431i	$-0.435327\pi$
$389$	3096.24	0.403562	0.201781	+	0.979431i	$0.435327\pi$
$390$	0	0
$391$	1453.90	0.188048
$392$	0	0
$393$	2826.42	0.362783
$394$	0	0
$395$	795.984	0.101393
$396$	0	0
$397$	1768.38	0.223558	0.111779	−	0.993733i	$-0.464345\pi$
$397$	1768.38	0.223558	0.111779	+	0.993733i	$0.464345\pi$
$398$	0	0
$399$	9311.37	1.16830
$400$	0	0
$401$	−8948.75	−1.11441	−0.557206	−	0.830374i	$-0.688127\pi$
$401$	−8948.75	−1.11441	−0.557206	+	0.830374i	$0.688127\pi$
$402$	0	0
$403$	−2943.85	−0.363879
$404$	0	0
$405$	266.612	0.0327112
$406$	0	0
$407$	1991.26	0.242514
$408$	0	0
$409$	−9195.18	−1.11167	−0.555834	−	0.831293i	$-0.687601\pi$
$409$	−9195.18	−1.11167	−0.555834	+	0.831293i	$0.687601\pi$
$410$	0	0
$411$	−2482.14	−0.297895
$412$	0	0
$413$	23026.0	2.74343
$414$	0	0
$415$	−4185.35	−0.495063
$416$	0	0
$417$	3745.46	0.439846
$418$	0	0
$419$	13288.5	1.54937	0.774683	−	0.632350i	$-0.217909\pi$
$419$	13288.5	1.54937	0.774683	+	0.632350i	$0.217909\pi$
$420$	0	0
$421$	−3626.33	−0.419802	−0.209901	−	0.977723i	$-0.567314\pi$
$421$	−3626.33	−0.419802	−0.209901	+	0.977723i	$0.567314\pi$
$422$	0	0
$423$	1407.65	0.161802
$424$	0	0
$425$	1170.31	0.133573
$426$	0	0
$427$	−11997.2	−1.35968
$428$	0	0
$429$	344.526	0.0387735
$430$	0	0
$431$	−10301.8	−1.15133	−0.575663	−	0.817687i	$-0.695256\pi$
$431$	−10301.8	−1.15133	−0.575663	+	0.817687i	$0.695256\pi$
$432$	0	0
$433$	−12941.0	−1.43627	−0.718135	−	0.695903i	$-0.755004\pi$
$433$	−12941.0	−1.43627	−0.718135	+	0.695903i	$0.755004\pi$
$434$	0	0
$435$	1686.82	0.185924
$436$	0	0
$437$	17013.3	1.86237
$438$	0	0
$439$	7529.77	0.818625	0.409313	−	0.912394i	$-0.365769\pi$
$439$	7529.77	0.818625	0.409313	+	0.912394i	$0.365769\pi$
$440$	0	0
$441$	2938.41	0.317289
$442$	0	0
$443$	2920.13	0.313181	0.156591	−	0.987664i	$-0.449950\pi$
$443$	2920.13	0.313181	0.156591	+	0.987664i	$0.449950\pi$
$444$	0	0
$445$	3571.34	0.380444
$446$	0	0
$447$	−668.808	−0.0707685
$448$	0	0
$449$	−11984.2	−1.25962	−0.629811	−	0.776749i	$-0.716867\pi$
$449$	−11984.2	−1.25962	−0.629811	+	0.776749i	$0.716867\pi$
$450$	0	0
$451$	2426.70	0.253368
$452$	0	0
$453$	6233.00	0.646472
$454$	0	0
$455$	1107.16	0.114076
$456$	0	0
$457$	−1603.39	−0.164121	−0.0820604	−	0.996627i	$-0.526150\pi$
$457$	−1603.39	−0.164121	−0.0820604	+	0.996627i	$0.526150\pi$
$458$	0	0
$459$	−276.777	−0.0281456
$460$	0	0
$461$	−14306.6	−1.44539	−0.722696	−	0.691166i	$-0.757097\pi$
$461$	−14306.6	−1.44539	−0.722696	+	0.691166i	$0.757097\pi$
$462$	0	0
$463$	−6957.94	−0.698408	−0.349204	−	0.937047i	$-0.613548\pi$
$463$	−6957.94	−0.698408	−0.349204	+	0.937047i	$0.613548\pi$
$464$	0	0
$465$	2236.08	0.223001
$466$	0	0
$467$	12177.6	1.20666	0.603330	−	0.797492i	$-0.293840\pi$
$467$	12177.6	1.20666	0.603330	+	0.797492i	$0.293840\pi$
$468$	0	0
$469$	11877.4	1.16940
$470$	0	0
$471$	828.604	0.0810617
$472$	0	0
$473$	982.755	0.0955331
$474$	0	0
$475$	13694.8	1.32287
$476$	0	0
$477$	−766.423	−0.0735684
$478$	0	0
$479$	−1999.79	−0.190757	−0.0953786	−	0.995441i	$-0.530406\pi$
$479$	−1999.79	−0.190757	−0.0953786	+	0.995441i	$0.530406\pi$
$480$	0	0
$481$	2930.32	0.277778
$482$	0	0
$483$	11009.3	1.03715
$484$	0	0
$485$	−263.002	−0.0246233
$486$	0	0
$487$	−10931.3	−1.01714	−0.508568	−	0.861022i	$-0.669825\pi$
$487$	−10931.3	−1.01714	−0.508568	+	0.861022i	$0.669825\pi$
$488$	0	0
$489$	9283.79	0.858543
$490$	0	0
$491$	9264.78	0.851556	0.425778	−	0.904828i	$-0.360001\pi$
$491$	9264.78	0.851556	0.425778	+	0.904828i	$0.360001\pi$
$492$	0	0
$493$	−1751.14	−0.159974
$494$	0	0
$495$	−261.694	−0.0237622
$496$	0	0
$497$	14515.5	1.31008
$498$	0	0
$499$	10145.8	0.910195	0.455098	−	0.890442i	$-0.349604\pi$
$499$	10145.8	0.910195	0.455098	+	0.890442i	$0.349604\pi$
$500$	0	0
$501$	−9718.55	−0.866652
$502$	0	0
$503$	−16397.7	−1.45356	−0.726778	−	0.686872i	$-0.758983\pi$
$503$	−16397.7	−1.45356	−0.726778	+	0.686872i	$0.758983\pi$
$504$	0	0
$505$	−644.231	−0.0567681
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	20153.5	1.75499	0.877494	−	0.479588i	$-0.159214\pi$
$509$	20153.5	1.75499	0.877494	+	0.479588i	$0.159214\pi$
$510$	0	0
$511$	−20308.7	−1.75813
$512$	0	0
$513$	−3238.80	−0.278746
$514$	0	0
$515$	2960.46	0.253308
$516$	0	0
$517$	−1381.68	−0.117536
$518$	0	0
$519$	−943.820	−0.0798248
$520$	0	0
$521$	13607.3	1.14423	0.572116	−	0.820173i	$-0.306123\pi$
$521$	13607.3	1.14423	0.572116	+	0.820173i	$0.306123\pi$
$522$	0	0
$523$	−15066.8	−1.25970	−0.629852	−	0.776715i	$-0.716884\pi$
$523$	−15066.8	−1.25970	−0.629852	+	0.776715i	$0.716884\pi$
$524$	0	0
$525$	8861.97	0.736701
$526$	0	0
$527$	−2321.33	−0.191876
$528$	0	0
$529$	7948.76	0.653305
$530$	0	0
$531$	−8009.20	−0.654557
$532$	0	0
$533$	3571.11	0.290210
$534$	0	0
$535$	1643.46	0.132809
$536$	0	0
$537$	8235.65	0.661815
$538$	0	0
$539$	−2884.21	−0.230486
$540$	0	0
$541$	20964.6	1.66606	0.833029	−	0.553229i	$-0.186605\pi$
$541$	20964.6	1.66606	0.833029	+	0.553229i	$0.186605\pi$
$542$	0	0
$543$	9783.43	0.773199
$544$	0	0
$545$	6427.71	0.505198
$546$	0	0
$547$	−4564.15	−0.356762	−0.178381	−	0.983961i	$-0.557086\pi$
$547$	−4564.15	−0.356762	−0.178381	+	0.983961i	$0.557086\pi$
$548$	0	0
$549$	4173.01	0.324408
$550$	0	0
$551$	−20491.5	−1.58434
$552$	0	0
$553$	−6257.23	−0.481166
$554$	0	0
$555$	−2225.80	−0.170234
$556$	0	0
$557$	−21787.2	−1.65737	−0.828684	−	0.559716i	$-0.810910\pi$
$557$	−21787.2	−1.65737	−0.828684	+	0.559716i	$0.810910\pi$
$558$	0	0
$559$	1446.21	0.109424
$560$	0	0
$561$	271.671	0.0204456
$562$	0	0
$563$	−22930.9	−1.71656	−0.858279	−	0.513183i	$-0.828466\pi$
$563$	−22930.9	−1.71656	−0.858279	+	0.513183i	$0.828466\pi$
$564$	0	0
$565$	4226.85	0.314734
$566$	0	0
$567$	−2095.84	−0.155232
$568$	0	0
$569$	3375.90	0.248726	0.124363	−	0.992237i	$-0.460311\pi$
$569$	3375.90	0.248726	0.124363	+	0.992237i	$0.460311\pi$
$570$	0	0
$571$	1027.69	0.0753196	0.0376598	−	0.999291i	$-0.488010\pi$
$571$	1027.69	0.0753196	0.0376598	+	0.999291i	$0.488010\pi$
$572$	0	0
$573$	−7296.97	−0.531999
$574$	0	0
$575$	16192.2	1.17437
$576$	0	0
$577$	−17617.5	−1.27110	−0.635551	−	0.772059i	$-0.719227\pi$
$577$	−17617.5	−1.27110	−0.635551	+	0.772059i	$0.719227\pi$
$578$	0	0
$579$	−2915.54	−0.209267
$580$	0	0
$581$	32901.1	2.34934
$582$	0	0
$583$	752.286	0.0534417
$584$	0	0
$585$	−385.106	−0.0272174
$586$	0	0
$587$	9481.32	0.666671	0.333336	−	0.942808i	$-0.391826\pi$
$587$	9481.32	0.666671	0.333336	+	0.942808i	$0.391826\pi$
$588$	0	0
$589$	−27163.9	−1.90029
$590$	0	0
$591$	5643.32	0.392784
$592$	0	0
$593$	−14005.6	−0.969882	−0.484941	−	0.874547i	$-0.661159\pi$
$593$	−14005.6	−0.969882	−0.484941	+	0.874547i	$0.661159\pi$
$594$	0	0
$595$	873.035	0.0601529
$596$	0	0
$597$	4676.63	0.320606
$598$	0	0
$599$	−12057.8	−0.822486	−0.411243	−	0.911526i	$-0.634905\pi$
$599$	−12057.8	−0.822486	−0.411243	+	0.911526i	$0.634905\pi$
$600$	0	0
$601$	1000.24	0.0678878	0.0339439	−	0.999424i	$-0.489193\pi$
$601$	1000.24	0.0678878	0.0339439	+	0.999424i	$0.489193\pi$
$602$	0	0
$603$	−4131.36	−0.279009
$604$	0	0
$605$	−4124.12	−0.277140
$606$	0	0
$607$	5254.38	0.351349	0.175674	−	0.984448i	$-0.443789\pi$
$607$	5254.38	0.351349	0.175674	+	0.984448i	$0.443789\pi$
$608$	0	0
$609$	−13260.1	−0.882311
$610$	0	0
$611$	−2033.27	−0.134627
$612$	0	0
$613$	−3556.66	−0.234343	−0.117171	−	0.993112i	$-0.537383\pi$
$613$	−3556.66	−0.234343	−0.117171	+	0.993112i	$0.537383\pi$
$614$	0	0
$615$	−2712.53	−0.177854
$616$	0	0
$617$	−10897.9	−0.711071	−0.355536	−	0.934663i	$-0.615702\pi$
$617$	−10897.9	−0.711071	−0.355536	+	0.934663i	$0.615702\pi$
$618$	0	0
$619$	−27333.5	−1.77484	−0.887419	−	0.460965i	$-0.847504\pi$
$619$	−27333.5	−1.77484	−0.887419	+	0.460965i	$0.847504\pi$
$620$	0	0
$621$	−3829.41	−0.247454
$622$	0	0
$623$	−28074.3	−1.80541
$624$	0	0
$625$	11679.6	0.747496
$626$	0	0
$627$	3179.06	0.202487
$628$	0	0
$629$	2310.67	0.146474
$630$	0	0
$631$	13213.3	0.833617	0.416808	−	0.908994i	$-0.363149\pi$
$631$	13213.3	0.833617	0.416808	+	0.908994i	$0.363149\pi$
$632$	0	0
$633$	−6682.74	−0.419613
$634$	0	0
$635$	−4600.51	−0.287505
$636$	0	0
$637$	−4244.37	−0.264000
$638$	0	0
$639$	−5048.96	−0.312573
$640$	0	0
$641$	−30291.7	−1.86654	−0.933268	−	0.359181i	$-0.883056\pi$
$641$	−30291.7	−1.86654	−0.933268	+	0.359181i	$0.883056\pi$
$642$	0	0
$643$	−9072.67	−0.556440	−0.278220	−	0.960517i	$-0.589744\pi$
$643$	−9072.67	−0.556440	−0.278220	+	0.960517i	$0.589744\pi$
$644$	0	0
$645$	−1098.51	−0.0670601
$646$	0	0
$647$	19487.9	1.18416	0.592078	−	0.805880i	$-0.298308\pi$
$647$	19487.9	1.18416	0.592078	+	0.805880i	$0.298308\pi$
$648$	0	0
$649$	7861.47	0.475485
$650$	0	0
$651$	−17577.8	−1.05826
$652$	0	0
$653$	−13423.9	−0.804465	−0.402233	−	0.915537i	$-0.631766\pi$
$653$	−13423.9	−0.804465	−0.402233	+	0.915537i	$0.631766\pi$
$654$	0	0
$655$	3101.05	0.184989
$656$	0	0
$657$	7064.02	0.419473
$658$	0	0
$659$	−32529.2	−1.92285	−0.961425	−	0.275067i	$-0.911300\pi$
$659$	−32529.2	−1.92285	−0.961425	+	0.275067i	$0.911300\pi$
$660$	0	0
$661$	19133.6	1.12589	0.562944	−	0.826495i	$-0.309669\pi$
$661$	19133.6	1.12589	0.562944	+	0.826495i	$0.309669\pi$
$662$	0	0
$663$	399.788	0.0234185
$664$	0	0
$665$	10216.1	0.595736
$666$	0	0
$667$	−24228.3	−1.40648
$668$	0	0
$669$	12093.6	0.698902
$670$	0	0
$671$	−4096.04	−0.235657
$672$	0	0
$673$	30936.5	1.77194	0.885970	−	0.463742i	$-0.153493\pi$
$673$	30936.5	1.77194	0.885970	+	0.463742i	$0.153493\pi$
$674$	0	0
$675$	−3082.48	−0.175770
$676$	0	0
$677$	−8960.22	−0.508670	−0.254335	−	0.967116i	$-0.581857\pi$
$677$	−8960.22	−0.508670	−0.254335	+	0.967116i	$0.581857\pi$
$678$	0	0
$679$	2067.46	0.116851
$680$	0	0
$681$	−17632.6	−0.992192
$682$	0	0
$683$	25216.5	1.41271	0.706355	−	0.707858i	$-0.250338\pi$
$683$	25216.5	1.41271	0.706355	+	0.707858i	$0.250338\pi$
$684$	0	0
$685$	−2723.32	−0.151902
$686$	0	0
$687$	−11406.3	−0.633446
$688$	0	0
$689$	1107.06	0.0612126
$690$	0	0
$691$	28467.8	1.56725	0.783624	−	0.621236i	$-0.213369\pi$
$691$	28467.8	1.56725	0.783624	+	0.621236i	$0.213369\pi$
$692$	0	0
$693$	2057.18	0.112764
$694$	0	0
$695$	4109.40	0.224285
$696$	0	0
$697$	2815.95	0.153030
$698$	0	0
$699$	−10659.0	−0.576766
$700$	0	0
$701$	−12679.7	−0.683175	−0.341587	−	0.939850i	$-0.610965\pi$
$701$	−12679.7	−0.683175	−0.341587	+	0.939850i	$0.610965\pi$
$702$	0	0
$703$	27039.1	1.45064
$704$	0	0
$705$	1544.42	0.0825055
$706$	0	0
$707$	5064.30	0.269395
$708$	0	0
$709$	8986.61	0.476021	0.238011	−	0.971263i	$-0.423505\pi$
$709$	8986.61	0.476021	0.238011	+	0.971263i	$0.423505\pi$
$710$	0	0
$711$	2176.47	0.114802
$712$	0	0
$713$	−32117.4	−1.68696
$714$	0	0
$715$	378.002	0.0197713
$716$	0	0
$717$	−3921.98	−0.204281
$718$	0	0
$719$	32611.8	1.69154	0.845769	−	0.533549i	$-0.179142\pi$
$719$	32611.8	1.69154	0.845769	+	0.533549i	$0.179142\pi$
$720$	0	0
$721$	−23272.2	−1.20208
$722$	0	0
$723$	−8362.44	−0.430156
$724$	0	0
$725$	−19502.5	−0.999043
$726$	0	0
$727$	−25550.2	−1.30344	−0.651722	−	0.758458i	$-0.725953\pi$
$727$	−25550.2	−1.30344	−0.651722	+	0.758458i	$0.725953\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	1140.39	0.0577003
$732$	0	0
$733$	−2278.67	−0.114822	−0.0574111	−	0.998351i	$-0.518285\pi$
$733$	−2278.67	−0.114822	−0.0574111	+	0.998351i	$0.518285\pi$
$734$	0	0
$735$	3223.93	0.161791
$736$	0	0
$737$	4055.16	0.202678
$738$	0	0
$739$	24392.5	1.21420	0.607099	−	0.794626i	$-0.292333\pi$
$739$	24392.5	1.21420	0.607099	+	0.794626i	$0.292333\pi$
$740$	0	0
$741$	4678.27	0.231930
$742$	0	0
$743$	15792.0	0.779750	0.389875	−	0.920868i	$-0.372518\pi$
$743$	15792.0	0.779750	0.389875	+	0.920868i	$0.372518\pi$
$744$	0	0
$745$	−733.794	−0.0360861
$746$	0	0
$747$	−11444.1	−0.560531
$748$	0	0
$749$	−12919.3	−0.630253
$750$	0	0
$751$	10793.7	0.524457	0.262228	−	0.965006i	$-0.415543\pi$
$751$	10793.7	0.524457	0.262228	+	0.965006i	$0.415543\pi$
$752$	0	0
$753$	−2182.15	−0.105607
$754$	0	0
$755$	6838.64	0.329647
$756$	0	0
$757$	21486.3	1.03162	0.515809	−	0.856704i	$-0.327491\pi$
$757$	21486.3	1.03162	0.515809	+	0.856704i	$0.327491\pi$
$758$	0	0
$759$	3758.78	0.179756
$760$	0	0
$761$	887.661	0.0422834	0.0211417	−	0.999776i	$-0.493270\pi$
$761$	887.661	0.0422834	0.0211417	+	0.999776i	$0.493270\pi$
$762$	0	0
$763$	−50528.2	−2.39744
$764$	0	0
$765$	−303.670	−0.0143519
$766$	0	0
$767$	11568.8	0.544624
$768$	0	0
$769$	10875.7	0.509997	0.254998	−	0.966941i	$-0.417925\pi$
$769$	10875.7	0.509997	0.254998	+	0.966941i	$0.417925\pi$
$770$	0	0
$771$	−6165.39	−0.287991
$772$	0	0
$773$	−8068.68	−0.375434	−0.187717	−	0.982223i	$-0.560109\pi$
$773$	−8068.68	−0.375434	−0.187717	+	0.982223i	$0.560109\pi$
$774$	0	0
$775$	−25852.9	−1.19827
$776$	0	0
$777$	17497.1	0.807855
$778$	0	0
$779$	32951.9	1.51556
$780$	0	0
$781$	4955.83	0.227060
$782$	0	0
$783$	4612.31	0.210511
$784$	0	0
$785$	909.118	0.0413348
$786$	0	0
$787$	12769.9	0.578397	0.289198	−	0.957269i	$-0.406611\pi$
$787$	12769.9	0.578397	0.289198	+	0.957269i	$0.406611\pi$
$788$	0	0
$789$	−2553.96	−0.115239
$790$	0	0
$791$	−33227.3	−1.49358
$792$	0	0
$793$	−6027.68	−0.269923
$794$	0	0
$795$	−840.895	−0.0375138
$796$	0	0
$797$	−17544.9	−0.779763	−0.389881	−	0.920865i	$-0.627484\pi$
$797$	−17544.9	−0.779763	−0.389881	+	0.920865i	$0.627484\pi$
$798$	0	0
$799$	−1603.31	−0.0709899
$800$	0	0
$801$	9765.15	0.430755
$802$	0	0
$803$	−6933.72	−0.304715
$804$	0	0
$805$	12079.1	0.528860
$806$	0	0
$807$	16794.4	0.732578
$808$	0	0
$809$	11082.8	0.481643	0.240822	−	0.970569i	$-0.422583\pi$
$809$	11082.8	0.481643	0.240822	+	0.970569i	$0.422583\pi$
$810$	0	0
$811$	−16057.5	−0.695259	−0.347629	−	0.937632i	$-0.613013\pi$
$811$	−16057.5	−0.695259	−0.347629	+	0.937632i	$0.613013\pi$
$812$	0	0
$813$	−18731.5	−0.808045
$814$	0	0
$815$	10185.9	0.437786
$816$	0	0
$817$	13344.7	0.571447
$818$	0	0
$819$	3027.32	0.129161
$820$	0	0
$821$	−8634.49	−0.367047	−0.183524	−	0.983015i	$-0.558750\pi$
$821$	−8634.49	−0.367047	−0.183524	+	0.983015i	$0.558750\pi$
$822$	0	0
$823$	38764.2	1.64184	0.820921	−	0.571041i	$-0.193460\pi$
$823$	38764.2	1.64184	0.820921	+	0.571041i	$0.193460\pi$
$824$	0	0
$825$	3025.62	0.127683
$826$	0	0
$827$	5820.03	0.244719	0.122359	−	0.992486i	$-0.460954\pi$
$827$	5820.03	0.244719	0.122359	+	0.992486i	$0.460954\pi$
$828$	0	0
$829$	30389.8	1.27320	0.636599	−	0.771195i	$-0.280341\pi$
$829$	30389.8	1.27320	0.636599	+	0.771195i	$0.280341\pi$
$830$	0	0
$831$	−16363.1	−0.683069
$832$	0	0
$833$	−3346.85	−0.139209
$834$	0	0
$835$	−10662.9	−0.441921
$836$	0	0
$837$	6114.14	0.252492
$838$	0	0
$839$	363.306	0.0149496	0.00747480	−	0.999972i	$-0.497621\pi$
$839$	363.306	0.0149496	0.00747480	+	0.999972i	$0.497621\pi$
$840$	0	0
$841$	4792.56	0.196505
$842$	0	0
$843$	−14241.3	−0.581845
$844$	0	0
$845$	556.264	0.0226462
$846$	0	0
$847$	32419.7	1.31518
$848$	0	0
$849$	−5251.61	−0.212291
$850$	0	0
$851$	31969.8	1.28779
$852$	0	0
$853$	−33784.6	−1.35611	−0.678055	−	0.735011i	$-0.737177\pi$
$853$	−33784.6	−1.35611	−0.678055	+	0.735011i	$0.737177\pi$
$854$	0	0
$855$	−3553.51	−0.142137
$856$	0	0
$857$	−30233.0	−1.20506	−0.602532	−	0.798095i	$-0.705841\pi$
$857$	−30233.0	−1.20506	−0.602532	+	0.798095i	$0.705841\pi$
$858$	0	0
$859$	44459.3	1.76593	0.882963	−	0.469443i	$-0.155545\pi$
$859$	44459.3	1.76593	0.882963	+	0.469443i	$0.155545\pi$
$860$	0	0
$861$	21323.2	0.844011
$862$	0	0
$863$	21795.6	0.859710	0.429855	−	0.902898i	$-0.358565\pi$
$863$	21795.6	0.859710	0.429855	+	0.902898i	$0.358565\pi$
$864$	0	0
$865$	−1035.53	−0.0407041
$866$	0	0
$867$	−14423.8	−0.565001
$868$	0	0
$869$	−2136.32	−0.0833945
$870$	0	0
$871$	5967.53	0.232149
$872$	0	0
$873$	−719.129	−0.0278795
$874$	0	0
$875$	20368.8	0.786962
$876$	0	0
$877$	−45759.9	−1.76192	−0.880959	−	0.473192i	$-0.843102\pi$
$877$	−45759.9	−1.76192	−0.880959	+	0.473192i	$0.843102\pi$
$878$	0	0
$879$	8552.30	0.328171
$880$	0	0
$881$	8258.13	0.315804	0.157902	−	0.987455i	$-0.449527\pi$
$881$	8258.13	0.315804	0.157902	+	0.987455i	$0.449527\pi$
$882$	0	0
$883$	48369.6	1.84345	0.921725	−	0.387844i	$-0.126780\pi$
$883$	48369.6	1.84345	0.921725	+	0.387844i	$0.126780\pi$
$884$	0	0
$885$	−8787.43	−0.333770
$886$	0	0
$887$	−31231.5	−1.18225	−0.591123	−	0.806582i	$-0.701315\pi$
$887$	−31231.5	−1.18225	−0.591123	+	0.806582i	$0.701315\pi$
$888$	0	0
$889$	36164.6	1.36437
$890$	0	0
$891$	−715.553	−0.0269045
$892$	0	0
$893$	−18761.7	−0.703063
$894$	0	0
$895$	9035.89	0.337471
$896$	0	0
$897$	5531.37	0.205894
$898$	0	0
$899$	38683.5	1.43511
$900$	0	0
$901$	872.955	0.0322778
$902$	0	0
$903$	8635.39	0.318237
$904$	0	0
$905$	10734.1	0.394268
$906$	0	0
$907$	2275.92	0.0833194	0.0416597	−	0.999132i	$-0.486735\pi$
$907$	2275.92	0.0833194	0.0416597	+	0.999132i	$0.486735\pi$
$908$	0	0
$909$	−1761.53	−0.0642753
$910$	0	0
$911$	−5892.87	−0.214314	−0.107157	−	0.994242i	$-0.534175\pi$
$911$	−5892.87	−0.214314	−0.107157	+	0.994242i	$0.534175\pi$
$912$	0	0
$913$	11233.0	0.407182
$914$	0	0
$915$	4578.49	0.165421
$916$	0	0
$917$	−24377.4	−0.877875
$918$	0	0
$919$	33385.3	1.19835	0.599174	−	0.800619i	$-0.295496\pi$
$919$	33385.3	1.19835	0.599174	+	0.800619i	$0.295496\pi$
$920$	0	0
$921$	−16671.1	−0.596450
$922$	0	0
$923$	7292.95	0.260076
$924$	0	0
$925$	25734.1	0.914736
$926$	0	0
$927$	8094.82	0.286806
$928$	0	0
$929$	41697.6	1.47261	0.736305	−	0.676650i	$-0.236569\pi$
$929$	41697.6	1.47261	0.736305	+	0.676650i	$0.236569\pi$
$930$	0	0
$931$	−39164.3	−1.37869
$932$	0	0
$933$	−14576.6	−0.511488
$934$	0	0
$935$	298.069	0.0104256
$936$	0	0
$937$	5508.64	0.192059	0.0960296	−	0.995378i	$-0.469386\pi$
$937$	5508.64	0.192059	0.0960296	+	0.995378i	$0.469386\pi$
$938$	0	0
$939$	−28847.3	−1.00255
$940$	0	0
$941$	−45425.2	−1.57367	−0.786833	−	0.617166i	$-0.788281\pi$
$941$	−45425.2	−1.57367	−0.786833	+	0.617166i	$0.788281\pi$
$942$	0	0
$943$	38960.8	1.34543
$944$	0	0
$945$	−2299.48	−0.0791557
$946$	0	0
$947$	20693.6	0.710086	0.355043	−	0.934850i	$-0.384466\pi$
$947$	20693.6	0.710086	0.355043	+	0.934850i	$0.384466\pi$
$948$	0	0
$949$	−10203.6	−0.349023
$950$	0	0
$951$	2346.28	0.0800036
$952$	0	0
$953$	−2163.05	−0.0735237	−0.0367619	−	0.999324i	$-0.511704\pi$
$953$	−2163.05	−0.0735237	−0.0367619	+	0.999324i	$0.511704\pi$
$954$	0	0
$955$	−8006.00	−0.271276
$956$	0	0
$957$	−4527.23	−0.152920
$958$	0	0
$959$	21408.1	0.720858
$960$	0	0
$961$	21488.4	0.721306
$962$	0	0
$963$	4493.74	0.150373
$964$	0	0
$965$	−3198.83	−0.106709
$966$	0	0
$967$	3163.84	0.105214	0.0526072	−	0.998615i	$-0.483247\pi$
$967$	3163.84	0.105214	0.0526072	+	0.998615i	$0.483247\pi$
$968$	0	0
$969$	3688.99	0.122299
$970$	0	0
$971$	7595.56	0.251033	0.125517	−	0.992092i	$-0.459941\pi$
$971$	7595.56	0.251033	0.125517	+	0.992092i	$0.459941\pi$
$972$	0	0
$973$	−32304.0	−1.06436
$974$	0	0
$975$	4452.47	0.146250
$976$	0	0
$977$	−49857.6	−1.63264	−0.816318	−	0.577602i	$-0.803989\pi$
$977$	−49857.6	−1.63264	−0.816318	+	0.577602i	$0.803989\pi$
$978$	0	0
$979$	−9585.03	−0.312910
$980$	0	0
$981$	17575.4	0.572006
$982$	0	0
$983$	−43762.5	−1.41995	−0.709973	−	0.704228i	$-0.751293\pi$
$983$	−43762.5	−1.41995	−0.709973	+	0.704228i	$0.751293\pi$
$984$	0	0
$985$	6191.66	0.200287
$986$	0	0
$987$	−12140.7	−0.391533
$988$	0	0
$989$	15778.2	0.507297
$990$	0	0
$991$	−36147.8	−1.15870	−0.579350	−	0.815079i	$-0.696693\pi$
$991$	−36147.8	−1.15870	−0.579350	+	0.815079i	$0.696693\pi$
$992$	0	0
$993$	−5123.01	−0.163720
$994$	0	0
$995$	5131.04	0.163482
$996$	0	0
$997$	26713.1	0.848558	0.424279	−	0.905531i	$-0.360528\pi$
$997$	26713.1	0.848558	0.424279	+	0.905531i	$0.360528\pi$
$998$	0	0
$999$	−6086.05	−0.192747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.e.1.2	✓	2
3.2	odd	2		936.4.a.f.1.1		2
4.3	odd	2		624.4.a.k.1.2		2
8.3	odd	2		2496.4.a.bh.1.1		2
8.5	even	2		2496.4.a.y.1.1		2
12.11	even	2		1872.4.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.e.1.2	✓	2	1.1	even	1	trivial
624.4.a.k.1.2		2	4.3	odd	2
936.4.a.f.1.1		2	3.2	odd	2
1872.4.a.bf.1.1		2	12.11	even	2
2496.4.a.y.1.1		2	8.5	even	2
2496.4.a.bh.1.1		2	8.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +3.29150 q^{5} -25.8745 q^{7} +9.00000 q^{9} -8.83399 q^{11} -13.0000 q^{13} +9.87451 q^{15} -10.2510 q^{17} -119.956 q^{19} -77.6235 q^{21} -141.830 q^{23} -114.166 q^{25} +27.0000 q^{27} +170.826 q^{29} +226.450 q^{31} -26.5020 q^{33} -85.1660 q^{35} -225.409 q^{37} -39.0000 q^{39} -274.701 q^{41} -111.247 q^{43} +29.6235 q^{45} +156.405 q^{47} +326.490 q^{49} -30.7530 q^{51} -85.1581 q^{53} -29.0771 q^{55} -359.867 q^{57} -889.911 q^{59} +463.668 q^{61} -232.871 q^{63} -42.7895 q^{65} -459.041 q^{67} -425.490 q^{69} -560.996 q^{71} +784.891 q^{73} -342.498 q^{75} +228.575 q^{77} +241.830 q^{79} +81.0000 q^{81} -1271.56 q^{83} -33.7411 q^{85} +512.478 q^{87} +1085.02 q^{89} +336.369 q^{91} +679.349 q^{93} -394.834 q^{95} -79.9032 q^{97} -79.5059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 4 q^{5} - 20 q^{7} + 18 q^{9} - 60 q^{11} - 26 q^{13} - 12 q^{15} - 84 q^{17} - 60 q^{19} - 60 q^{21} - 72 q^{23} - 186 q^{25} + 54 q^{27} - 124 q^{29} - 108 q^{31} - 180 q^{33} - 128 q^{35}+ \cdots - 540 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 4 * q^5 - 20 * q^7 + 18 * q^9 - 60 * q^11 - 26 * q^13 - 12 * q^15 - 84 * q^17 - 60 * q^19 - 60 * q^21 - 72 * q^23 - 186 * q^25 + 54 * q^27 - 124 * q^29 - 108 * q^31 - 180 * q^33 - 128 * q^35 + 36 * q^37 - 78 * q^39 - 52 * q^41 - 32 * q^43 - 36 * q^45 - 428 * q^47 + 18 * q^49 - 252 * q^51 + 380 * q^53 + 344 * q^55 - 180 * q^57 - 1420 * q^59 + 1012 * q^61 - 180 * q^63 + 52 * q^65 - 844 * q^67 - 216 * q^69 - 868 * q^71 - 60 * q^73 - 558 * q^75 - 72 * q^77 + 272 * q^79 + 162 * q^81 - 1252 * q^83 + 504 * q^85 - 372 * q^87 + 572 * q^89 + 260 * q^91 - 324 * q^93 - 832 * q^95 + 708 * q^97 - 540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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