LMFDB - Embedded newform 1881.4.a.n.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,4,Mod(1,1881)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1881.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1881.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:	$110.982592721$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	1881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.60095 q^{2} -1.23504 q^{4} -11.0151 q^{5} -27.4794 q^{7} +24.0199 q^{8} +28.6498 q^{10} +11.0000 q^{11} -78.9689 q^{13} +71.4727 q^{14} -52.5943 q^{16} +66.9785 q^{17} -19.0000 q^{19} +13.6041 q^{20} -28.6105 q^{22} -132.733 q^{23} -3.66746 q^{25} +205.394 q^{26} +33.9382 q^{28} +47.5040 q^{29} +267.324 q^{31} -55.3639 q^{32} -174.208 q^{34} +302.689 q^{35} +223.716 q^{37} +49.4181 q^{38} -264.582 q^{40} -420.576 q^{41} -238.252 q^{43} -13.5855 q^{44} +345.233 q^{46} +458.566 q^{47} +412.118 q^{49} +9.53889 q^{50} +97.5298 q^{52} -53.4502 q^{53} -121.166 q^{55} -660.053 q^{56} -123.556 q^{58} -415.991 q^{59} +515.513 q^{61} -695.298 q^{62} +564.754 q^{64} +869.851 q^{65} +1002.64 q^{67} -82.7212 q^{68} -787.279 q^{70} +433.124 q^{71} +594.021 q^{73} -581.875 q^{74} +23.4658 q^{76} -302.273 q^{77} +792.213 q^{79} +579.332 q^{80} +1093.90 q^{82} +177.134 q^{83} -737.775 q^{85} +619.682 q^{86} +264.219 q^{88} -606.280 q^{89} +2170.02 q^{91} +163.931 q^{92} -1192.71 q^{94} +209.287 q^{95} -1397.99 q^{97} -1071.90 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 4 q^{2} + 96 q^{4} - 14 q^{7} - 48 q^{8} - 124 q^{10} + 253 q^{11} - 150 q^{13} - 152 q^{14} + 444 q^{16} + 68 q^{17} - 437 q^{19} - 80 q^{20} - 44 q^{22} - 414 q^{23} + 383 q^{25} - 464 q^{26} - 384 q^{28}+ \cdots - 7876 q^{98}+O(q^{100}) $ 23 * q - 4 * q^2 + 96 * q^4 - 14 * q^7 - 48 * q^8 - 124 * q^10 + 253 * q^11 - 150 * q^13 - 152 * q^14 + 444 * q^16 + 68 * q^17 - 437 * q^19 - 80 * q^20 - 44 * q^22 - 414 * q^23 + 383 * q^25 - 464 * q^26 - 384 * q^28 - 604 * q^29 + 136 * q^31 - 448 * q^32 - 114 * q^34 - 312 * q^35 - 134 * q^37 + 76 * q^38 - 1518 * q^40 - 540 * q^41 - 742 * q^43 + 1056 * q^44 + 230 * q^46 - 426 * q^47 + 1059 * q^49 - 1322 * q^50 - 320 * q^52 - 1164 * q^53 - 5058 * q^56 + 2572 * q^58 - 1490 * q^59 - 666 * q^61 - 5010 * q^62 + 5582 * q^64 - 2836 * q^65 + 324 * q^67 + 656 * q^68 + 3226 * q^70 - 4188 * q^71 - 242 * q^73 - 4730 * q^74 - 1824 * q^76 - 154 * q^77 - 436 * q^79 - 4914 * q^80 - 52 * q^82 - 1322 * q^83 - 3650 * q^85 - 3298 * q^86 - 528 * q^88 - 1630 * q^89 + 876 * q^91 - 6110 * q^92 + 2494 * q^94 - 5162 * q^97 - 7876 * q^98

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 $

$2$	−2.60095	−0.919576	−0.459788	−	0.888029i	$-0.652075\pi$
$2$	−2.60095	−0.919576	−0.459788	+	0.888029i	$0.652075\pi$
$3$	0	0
$3$	0	0
$4$	−1.23504	−0.154380
$5$	−11.0151	−0.985221	−0.492610	−	0.870250i	$-0.663957\pi$
$5$	−11.0151	−0.985221	−0.492610	+	0.870250i	$0.663957\pi$
$6$	0	0
$7$	−27.4794	−1.48375	−0.741874	−	0.670539i	$-0.766063\pi$
$7$	−27.4794	−1.48375	−0.741874	+	0.670539i	$0.766063\pi$
$8$	24.0199	1.06154
$9$	0	0
$10$	28.6498	0.905985
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−78.9689	−1.68477	−0.842386	−	0.538875i	$-0.818849\pi$
$13$	−78.9689	−1.68477	−0.842386	+	0.538875i	$0.818849\pi$
$14$	71.4727	1.36442
$15$	0	0
$16$	−52.5943	−0.821787
$17$	66.9785	0.955569	0.477784	−	0.878477i	$-0.341440\pi$
$17$	66.9785	0.955569	0.477784	+	0.878477i	$0.341440\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	13.6041	0.152099
$21$	0	0
$22$	−28.6105	−0.277263
$23$	−132.733	−1.20334	−0.601669	−	0.798746i	$-0.705497\pi$
$23$	−132.733	−1.20334	−0.601669	+	0.798746i	$0.705497\pi$
$24$	0	0
$25$	−3.66746	−0.0293397
$26$	205.394	1.54928
$27$	0	0
$28$	33.9382	0.229061
$29$	47.5040	0.304182	0.152091	−	0.988366i	$-0.451399\pi$
$29$	47.5040	0.304182	0.152091	+	0.988366i	$0.451399\pi$
$30$	0	0
$31$	267.324	1.54880	0.774401	−	0.632695i	$-0.218051\pi$
$31$	267.324	1.54880	0.774401	+	0.632695i	$0.218051\pi$
$32$	−55.3639	−0.305845
$33$	0	0
$34$	−174.208	−0.878718
$35$	302.689	1.46182
$36$	0	0
$37$	223.716	0.994019	0.497009	−	0.867745i	$-0.334431\pi$
$37$	223.716	0.994019	0.497009	+	0.867745i	$0.334431\pi$
$38$	49.4181	0.210965
$39$	0	0
$40$	−264.582	−1.04585
$41$	−420.576	−1.60202	−0.801011	−	0.598650i	$-0.795704\pi$
$41$	−420.576	−1.60202	−0.801011	+	0.598650i	$0.795704\pi$
$42$	0	0
$43$	−238.252	−0.844955	−0.422478	−	0.906373i	$-0.638839\pi$
$43$	−238.252	−0.844955	−0.422478	+	0.906373i	$0.638839\pi$
$44$	−13.5855	−0.0465474
$45$	0	0
$46$	345.233	1.10656
$47$	458.566	1.42316	0.711582	−	0.702603i	$-0.247979\pi$
$47$	458.566	1.42316	0.711582	+	0.702603i	$0.247979\pi$
$48$	0	0
$49$	412.118	1.20151
$50$	9.53889	0.0269801
$51$	0	0
$52$	97.5298	0.260095
$53$	−53.4502	−0.138527	−0.0692636	−	0.997598i	$-0.522065\pi$
$53$	−53.4502	−0.138527	−0.0692636	+	0.997598i	$0.522065\pi$
$54$	0	0
$55$	−121.166	−0.297055
$56$	−660.053	−1.57506
$57$	0	0
$58$	−123.556	−0.279718
$59$	−415.991	−0.917923	−0.458961	−	0.888456i	$-0.651778\pi$
$59$	−415.991	−0.917923	−0.458961	+	0.888456i	$0.651778\pi$
$60$	0	0
$61$	515.513	1.08204	0.541022	−	0.841009i	$-0.318038\pi$
$61$	515.513	1.08204	0.541022	+	0.841009i	$0.318038\pi$
$62$	−695.298	−1.42424
$63$	0	0
$64$	564.754	1.10303
$65$	869.851	1.65987
$66$	0	0
$67$	1002.64	1.82824	0.914120	−	0.405443i	$-0.132883\pi$
$67$	1002.64	1.82824	0.914120	+	0.405443i	$0.132883\pi$
$68$	−82.7212	−0.147521
$69$	0	0
$70$	−787.279	−1.34425
$71$	433.124	0.723977	0.361988	−	0.932183i	$-0.382098\pi$
$71$	433.124	0.723977	0.361988	+	0.932183i	$0.382098\pi$
$72$	0	0
$73$	594.021	0.952396	0.476198	−	0.879338i	$-0.342015\pi$
$73$	594.021	0.952396	0.476198	+	0.879338i	$0.342015\pi$
$74$	−581.875	−0.914076
$75$	0	0
$76$	23.4658	0.0354172
$77$	−302.273	−0.447367
$78$	0	0
$79$	792.213	1.12824	0.564119	−	0.825693i	$-0.309216\pi$
$79$	792.213	1.12824	0.564119	+	0.825693i	$0.309216\pi$
$80$	579.332	0.809641
$81$	0	0
$82$	1093.90	1.47318
$83$	177.134	0.234253	0.117126	−	0.993117i	$-0.462632\pi$
$83$	177.134	0.234253	0.117126	+	0.993117i	$0.462632\pi$
$84$	0	0
$85$	−737.775	−0.941446
$86$	619.682	0.777000
$87$	0	0
$88$	264.219	0.320066
$89$	−606.280	−0.722085	−0.361043	−	0.932549i	$-0.617579\pi$
$89$	−606.280	−0.722085	−0.361043	+	0.932549i	$0.617579\pi$
$90$	0	0
$91$	2170.02	2.49978
$92$	163.931	0.185771
$93$	0	0
$94$	−1192.71	−1.30871
$95$	209.287	0.226025
$96$	0	0
$97$	−1397.99	−1.46334	−0.731670	−	0.681659i	$-0.761259\pi$
$97$	−1397.99	−1.46334	−0.731670	+	0.681659i	$0.761259\pi$
$98$	−1071.90	−1.10488
$99$	0	0
$100$	4.52946	0.00452946
$101$	564.244	0.555885	0.277942	−	0.960598i	$-0.410348\pi$
$101$	564.244	0.555885	0.277942	+	0.960598i	$0.410348\pi$
$102$	0	0
$103$	−662.713	−0.633971	−0.316986	−	0.948430i	$-0.602671\pi$
$103$	−662.713	−0.633971	−0.316986	+	0.948430i	$0.602671\pi$
$104$	−1896.83	−1.78845
$105$	0	0
$106$	139.021	0.127386
$107$	1705.87	1.54124	0.770618	−	0.637298i	$-0.219948\pi$
$107$	1705.87	1.54124	0.770618	+	0.637298i	$0.219948\pi$
$108$	0	0
$109$	−112.839	−0.0991558	−0.0495779	−	0.998770i	$-0.515788\pi$
$109$	−112.839	−0.0991558	−0.0495779	+	0.998770i	$0.515788\pi$
$110$	315.148	0.273165
$111$	0	0
$112$	1445.26	1.21932
$113$	1276.36	1.06256	0.531282	−	0.847195i	$-0.321710\pi$
$113$	1276.36	1.06256	0.531282	+	0.847195i	$0.321710\pi$
$114$	0	0
$115$	1462.07	1.18555
$116$	−58.6694	−0.0469597
$117$	0	0
$118$	1081.97	0.844100
$119$	−1840.53	−1.41782
$120$	0	0
$121$	121.000	0.0909091
$122$	−1340.82	−0.995021
$123$	0	0
$124$	−330.157	−0.239104
$125$	1417.29	1.01413
$126$	0	0
$127$	−1702.15	−1.18930	−0.594652	−	0.803983i	$-0.702710\pi$
$127$	−1702.15	−1.18930	−0.594652	+	0.803983i	$0.702710\pi$
$128$	−1025.99	−0.708479
$129$	0	0
$130$	−2262.44	−1.52638
$131$	−1035.55	−0.690659	−0.345330	−	0.938481i	$-0.612233\pi$
$131$	−1035.55	−0.690659	−0.345330	+	0.938481i	$0.612233\pi$
$132$	0	0
$133$	522.109	0.340395
$134$	−2607.82	−1.68121
$135$	0	0
$136$	1608.82	1.01437
$137$	1476.92	0.921033	0.460516	−	0.887651i	$-0.347664\pi$
$137$	1476.92	0.921033	0.460516	+	0.887651i	$0.347664\pi$
$138$	0	0
$139$	932.490	0.569012	0.284506	−	0.958674i	$-0.408170\pi$
$139$	932.490	0.569012	0.284506	+	0.958674i	$0.408170\pi$
$140$	−373.833	−0.225676
$141$	0	0
$142$	−1126.53	−0.665751
$143$	−868.658	−0.507978
$144$	0	0
$145$	−523.262	−0.299687
$146$	−1545.02	−0.875800
$147$	0	0
$148$	−276.299	−0.153457
$149$	−3085.69	−1.69658	−0.848288	−	0.529536i	$-0.822366\pi$
$149$	−3085.69	−1.69658	−0.848288	+	0.529536i	$0.822366\pi$
$150$	0	0
$151$	−2129.53	−1.14767	−0.573837	−	0.818969i	$-0.694546\pi$
$151$	−2129.53	−1.14767	−0.573837	+	0.818969i	$0.694546\pi$
$152$	−456.378	−0.243534
$153$	0	0
$154$	786.199	0.411388
$155$	−2944.61	−1.52591
$156$	0	0
$157$	710.284	0.361063	0.180531	−	0.983569i	$-0.442218\pi$
$157$	710.284	0.361063	0.180531	+	0.983569i	$0.442218\pi$
$158$	−2060.51	−1.03750
$159$	0	0
$160$	609.839	0.301325
$161$	3647.43	1.78545
$162$	0	0
$163$	−84.4587	−0.0405848	−0.0202924	−	0.999794i	$-0.506460\pi$
$163$	−84.4587	−0.0405848	−0.0202924	+	0.999794i	$0.506460\pi$
$164$	519.428	0.247320
$165$	0	0
$166$	−460.717	−0.215413
$167$	1232.58	0.571136	0.285568	−	0.958358i	$-0.407818\pi$
$167$	1232.58	0.571136	0.285568	+	0.958358i	$0.407818\pi$
$168$	0	0
$169$	4039.08	1.83845
$170$	1918.92	0.865731
$171$	0	0
$172$	294.251	0.130444
$173$	174.321	0.0766091	0.0383045	−	0.999266i	$-0.487804\pi$
$173$	174.321	0.0766091	0.0383045	+	0.999266i	$0.487804\pi$
$174$	0	0
$175$	100.780	0.0435327
$176$	−578.538	−0.247778
$177$	0	0
$178$	1576.91	0.664012
$179$	737.610	0.307997	0.153999	−	0.988071i	$-0.450785\pi$
$179$	737.610	0.307997	0.153999	+	0.988071i	$0.450785\pi$
$180$	0	0
$181$	−1572.89	−0.645921	−0.322960	−	0.946413i	$-0.604678\pi$
$181$	−1572.89	−0.645921	−0.322960	+	0.946413i	$0.604678\pi$
$182$	−5644.12	−2.29873
$183$	0	0
$184$	−3188.24	−1.27739
$185$	−2464.26	−0.979328
$186$	0	0
$187$	736.763	0.288115
$188$	−566.348	−0.219708
$189$	0	0
$190$	−544.346	−0.207847
$191$	−2043.11	−0.774003	−0.387002	−	0.922079i	$-0.626489\pi$
$191$	−2043.11	−0.774003	−0.387002	+	0.922079i	$0.626489\pi$
$192$	0	0
$193$	3582.59	1.33617	0.668084	−	0.744086i	$-0.267115\pi$
$193$	3582.59	1.33617	0.668084	+	0.744086i	$0.267115\pi$
$194$	3636.10	1.34565
$195$	0	0
$196$	−508.982	−0.185489
$197$	−4358.87	−1.57643	−0.788214	−	0.615401i	$-0.788994\pi$
$197$	−4358.87	−1.57643	−0.788214	+	0.615401i	$0.788994\pi$
$198$	0	0
$199$	5360.89	1.90967	0.954833	−	0.297143i	$-0.0960338\pi$
$199$	5360.89	1.90967	0.954833	+	0.297143i	$0.0960338\pi$
$200$	−88.0920	−0.0311452
$201$	0	0
$202$	−1467.57	−0.511178
$203$	−1305.38	−0.451330
$204$	0	0
$205$	4632.68	1.57834
$206$	1723.68	0.582984
$207$	0	0
$208$	4153.32	1.38452
$209$	−209.000	−0.0691714
$210$	0	0
$211$	−4155.90	−1.35594	−0.677972	−	0.735088i	$-0.737141\pi$
$211$	−4155.90	−1.35594	−0.677972	+	0.735088i	$0.737141\pi$
$212$	66.0131	0.0213859
$213$	0	0
$214$	−4436.88	−1.41728
$215$	2624.37	0.832467
$216$	0	0
$217$	−7345.91	−2.29803
$218$	293.488	0.0911813
$219$	0	0
$220$	149.645	0.0458594
$221$	−5289.21	−1.60991
$222$	0	0
$223$	−624.537	−0.187543	−0.0937715	−	0.995594i	$-0.529892\pi$
$223$	−624.537	−0.187543	−0.0937715	+	0.995594i	$0.529892\pi$
$224$	1521.37	0.453797
$225$	0	0
$226$	−3319.75	−0.977108
$227$	3437.57	1.00511	0.502554	−	0.864546i	$-0.332394\pi$
$227$	3437.57	1.00511	0.502554	+	0.864546i	$0.332394\pi$
$228$	0	0
$229$	−2399.95	−0.692548	−0.346274	−	0.938133i	$-0.612553\pi$
$229$	−2399.95	−0.692548	−0.346274	+	0.938133i	$0.612553\pi$
$230$	−3802.77	−1.09021
$231$	0	0
$232$	1141.04	0.322901
$233$	1045.80	0.294045	0.147022	−	0.989133i	$-0.453031\pi$
$233$	1045.80	0.294045	0.147022	+	0.989133i	$0.453031\pi$
$234$	0	0
$235$	−5051.15	−1.40213
$236$	513.766	0.141709
$237$	0	0
$238$	4787.13	1.30380
$239$	−6954.65	−1.88225	−0.941127	−	0.338054i	$-0.890231\pi$
$239$	−6954.65	−1.88225	−0.941127	+	0.338054i	$0.890231\pi$
$240$	0	0
$241$	4832.51	1.29166	0.645828	−	0.763483i	$-0.276512\pi$
$241$	4832.51	1.29166	0.645828	+	0.763483i	$0.276512\pi$
$242$	−314.715	−0.0835978
$243$	0	0
$244$	−636.679	−0.167046
$245$	−4539.52	−1.18375
$246$	0	0
$247$	1500.41	0.386513
$248$	6421.11	1.64412
$249$	0	0
$250$	−3686.29	−0.932567
$251$	−965.793	−0.242870	−0.121435	−	0.992599i	$-0.538750\pi$
$251$	−965.793	−0.242870	−0.121435	+	0.992599i	$0.538750\pi$
$252$	0	0
$253$	−1460.06	−0.362820
$254$	4427.22	1.09366
$255$	0	0
$256$	−1849.48	−0.451534
$257$	2548.96	0.618676	0.309338	−	0.950952i	$-0.399893\pi$
$257$	2548.96	0.618676	0.309338	+	0.950952i	$0.399893\pi$
$258$	0	0
$259$	−6147.58	−1.47487
$260$	−1074.30	−0.256251
$261$	0	0
$262$	2693.42	0.635114
$263$	2008.18	0.470834	0.235417	−	0.971894i	$-0.424354\pi$
$263$	2008.18	0.470834	0.235417	+	0.971894i	$0.424354\pi$
$264$	0	0
$265$	588.759	0.136480
$266$	−1357.98	−0.313019
$267$	0	0
$268$	−1238.30	−0.282244
$269$	−6822.01	−1.54626	−0.773132	−	0.634245i	$-0.781311\pi$
$269$	−6822.01	−1.54626	−0.773132	+	0.634245i	$0.781311\pi$
$270$	0	0
$271$	−3430.15	−0.768882	−0.384441	−	0.923150i	$-0.625606\pi$
$271$	−3430.15	−0.768882	−0.384441	+	0.923150i	$0.625606\pi$
$272$	−3522.69	−0.785273
$273$	0	0
$274$	−3841.39	−0.846959
$275$	−40.3421	−0.00884624
$276$	0	0
$277$	2708.42	0.587484	0.293742	−	0.955885i	$-0.405099\pi$
$277$	2708.42	0.587484	0.293742	+	0.955885i	$0.405099\pi$
$278$	−2425.36	−0.523250
$279$	0	0
$280$	7270.55	1.55178
$281$	−2540.88	−0.539418	−0.269709	−	0.962942i	$-0.586927\pi$
$281$	−2540.88	−0.539418	−0.269709	+	0.962942i	$0.586927\pi$
$282$	0	0
$283$	2104.29	0.442003	0.221002	−	0.975273i	$-0.429067\pi$
$283$	2104.29	0.442003	0.221002	+	0.975273i	$0.429067\pi$
$284$	−534.926	−0.111768
$285$	0	0
$286$	2259.34	0.467124
$287$	11557.2	2.37700
$288$	0	0
$289$	−426.884	−0.0868887
$290$	1360.98	0.275585
$291$	0	0
$292$	−733.640	−0.147031
$293$	7324.22	1.46036	0.730180	−	0.683255i	$-0.239436\pi$
$293$	7324.22	1.46036	0.730180	+	0.683255i	$0.239436\pi$
$294$	0	0
$295$	4582.19	0.904357
$296$	5373.64	1.05519
$297$	0	0
$298$	8025.74	1.56013
$299$	10481.8	2.02735
$300$	0	0
$301$	6547.02	1.25370
$302$	5538.82	1.05537
$303$	0	0
$304$	999.293	0.188531
$305$	−5678.43	−1.06605
$306$	0	0
$307$	1203.41	0.223721	0.111860	−	0.993724i	$-0.464319\pi$
$307$	1203.41	0.223721	0.111860	+	0.993724i	$0.464319\pi$
$308$	373.320	0.0690646
$309$	0	0
$310$	7658.78	1.40319
$311$	−1120.80	−0.204355	−0.102178	−	0.994766i	$-0.532581\pi$
$311$	−1120.80	−0.204355	−0.102178	+	0.994766i	$0.532581\pi$
$312$	0	0
$313$	−5041.71	−0.910460	−0.455230	−	0.890374i	$-0.650443\pi$
$313$	−5041.71	−0.910460	−0.455230	+	0.890374i	$0.650443\pi$
$314$	−1847.42	−0.332024
$315$	0	0
$316$	−978.415	−0.174178
$317$	−1993.25	−0.353161	−0.176581	−	0.984286i	$-0.556504\pi$
$317$	−1993.25	−0.353161	−0.176581	+	0.984286i	$0.556504\pi$
$318$	0	0
$319$	522.544	0.0917143
$320$	−6220.82	−1.08673
$321$	0	0
$322$	−9486.79	−1.64186
$323$	−1272.59	−0.219222
$324$	0	0
$325$	289.615	0.0494306
$326$	219.673	0.0373208
$327$	0	0
$328$	−10102.2	−1.70061
$329$	−12601.1	−2.11162
$330$	0	0
$331$	8002.34	1.32885	0.664424	−	0.747356i	$-0.268677\pi$
$331$	8002.34	1.32885	0.664424	+	0.747356i	$0.268677\pi$
$332$	−218.768	−0.0361640
$333$	0	0
$334$	−3205.88	−0.525203
$335$	−11044.2	−1.80122
$336$	0	0
$337$	2804.64	0.453348	0.226674	−	0.973971i	$-0.427215\pi$
$337$	2804.64	0.453348	0.226674	+	0.973971i	$0.427215\pi$
$338$	−10505.5	−1.69060
$339$	0	0
$340$	911.182	0.145341
$341$	2940.57	0.466981
$342$	0	0
$343$	−1899.32	−0.298989
$344$	−5722.79	−0.896954
$345$	0	0
$346$	−453.400	−0.0704478
$347$	−11414.7	−1.76592	−0.882962	−	0.469444i	$-0.844454\pi$
$347$	−11414.7	−1.76592	−0.882962	+	0.469444i	$0.844454\pi$
$348$	0	0
$349$	−7543.53	−1.15701	−0.578504	−	0.815679i	$-0.696363\pi$
$349$	−7543.53	−1.15701	−0.578504	+	0.815679i	$0.696363\pi$
$350$	−262.123	−0.0400316
$351$	0	0
$352$	−609.002	−0.0922157
$353$	−9814.49	−1.47981	−0.739904	−	0.672712i	$-0.765129\pi$
$353$	−9814.49	−1.47981	−0.739904	+	0.672712i	$0.765129\pi$
$354$	0	0
$355$	−4770.90	−0.713277
$356$	748.781	0.111476
$357$	0	0
$358$	−1918.49	−0.283227
$359$	8743.57	1.28543	0.642713	−	0.766107i	$-0.277809\pi$
$359$	8743.57	1.28543	0.642713	+	0.766107i	$0.277809\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	4091.00	0.593973
$363$	0	0
$364$	−2680.06	−0.385916
$365$	−6543.20	−0.938321
$366$	0	0
$367$	9437.43	1.34232	0.671158	−	0.741315i	$-0.265797\pi$
$367$	9437.43	1.34232	0.671158	+	0.741315i	$0.265797\pi$
$368$	6981.01	0.988887
$369$	0	0
$370$	6409.41	0.900566
$371$	1468.78	0.205540
$372$	0	0
$373$	8848.72	1.22834	0.614168	−	0.789175i	$-0.289492\pi$
$373$	8848.72	1.22834	0.614168	+	0.789175i	$0.289492\pi$
$374$	−1916.29	−0.264943
$375$	0	0
$376$	11014.7	1.51075
$377$	−3751.34	−0.512477
$378$	0	0
$379$	6308.87	0.855052	0.427526	−	0.904003i	$-0.359385\pi$
$379$	6308.87	0.855052	0.427526	+	0.904003i	$0.359385\pi$
$380$	−258.478	−0.0348938
$381$	0	0
$382$	5314.05	0.711755
$383$	−8737.78	−1.16574	−0.582872	−	0.812564i	$-0.698071\pi$
$383$	−8737.78	−1.16574	−0.582872	+	0.812564i	$0.698071\pi$
$384$	0	0
$385$	3329.57	0.440755
$386$	−9318.14	−1.22871
$387$	0	0
$388$	1726.57	0.225911
$389$	−3317.96	−0.432461	−0.216230	−	0.976342i	$-0.569376\pi$
$389$	−3317.96	−0.432461	−0.216230	+	0.976342i	$0.569376\pi$
$390$	0	0
$391$	−8890.26	−1.14987
$392$	9899.03	1.27545
$393$	0	0
$394$	11337.2	1.44965
$395$	−8726.31	−1.11156
$396$	0	0
$397$	−5250.60	−0.663778	−0.331889	−	0.943318i	$-0.607686\pi$
$397$	−5250.60	−0.663778	−0.331889	+	0.943318i	$0.607686\pi$
$398$	−13943.4	−1.75608
$399$	0	0
$400$	192.888	0.0241110
$401$	−3117.56	−0.388239	−0.194119	−	0.980978i	$-0.562185\pi$
$401$	−3117.56	−0.388239	−0.194119	+	0.980978i	$0.562185\pi$
$402$	0	0
$403$	−21110.3	−2.60938
$404$	−696.864	−0.0858175
$405$	0	0
$406$	3395.24	0.415032
$407$	2460.88	0.299708
$408$	0	0
$409$	−2899.10	−0.350492	−0.175246	−	0.984525i	$-0.556072\pi$
$409$	−2899.10	−0.350492	−0.175246	+	0.984525i	$0.556072\pi$
$410$	−12049.4	−1.45141
$411$	0	0
$412$	818.477	0.0978725
$413$	11431.2	1.36197
$414$	0	0
$415$	−1951.15	−0.230791
$416$	4372.02	0.515279
$417$	0	0
$418$	543.599	0.0636084
$419$	−6516.51	−0.759791	−0.379896	−	0.925029i	$-0.624040\pi$
$419$	−6516.51	−0.759791	−0.379896	+	0.925029i	$0.624040\pi$
$420$	0	0
$421$	−12366.5	−1.43160	−0.715802	−	0.698304i	$-0.753939\pi$
$421$	−12366.5	−1.43160	−0.715802	+	0.698304i	$0.753939\pi$
$422$	10809.3	1.24689
$423$	0	0
$424$	−1283.87	−0.147052
$425$	−245.641	−0.0280361
$426$	0	0
$427$	−14166.0	−1.60548
$428$	−2106.81	−0.237936
$429$	0	0
$430$	−6825.86	−0.765517
$431$	−17643.1	−1.97178	−0.985891	−	0.167391i	$-0.946466\pi$
$431$	−17643.1	−1.97178	−0.985891	+	0.167391i	$0.946466\pi$
$432$	0	0
$433$	14644.6	1.62535	0.812673	−	0.582720i	$-0.198012\pi$
$433$	14644.6	1.62535	0.812673	+	0.582720i	$0.198012\pi$
$434$	19106.4	2.11322
$435$	0	0
$436$	139.360	0.0153077
$437$	2521.93	0.276065
$438$	0	0
$439$	−16663.0	−1.81158	−0.905790	−	0.423728i	$-0.860721\pi$
$439$	−16663.0	−1.81158	−0.905790	+	0.423728i	$0.860721\pi$
$440$	−2910.40	−0.315336
$441$	0	0
$442$	13757.0	1.48044
$443$	−11348.9	−1.21716	−0.608582	−	0.793491i	$-0.708262\pi$
$443$	−11348.9	−1.21716	−0.608582	+	0.793491i	$0.708262\pi$
$444$	0	0
$445$	6678.24	0.711414
$446$	1624.39	0.172460
$447$	0	0
$448$	−15519.1	−1.63663
$449$	1227.01	0.128967	0.0644834	−	0.997919i	$-0.479460\pi$
$449$	1227.01	0.128967	0.0644834	+	0.997919i	$0.479460\pi$
$450$	0	0
$451$	−4626.33	−0.483028
$452$	−1576.36	−0.164039
$453$	0	0
$454$	−8940.96	−0.924274
$455$	−23903.0	−2.46283
$456$	0	0
$457$	−15324.5	−1.56860	−0.784302	−	0.620380i	$-0.786978\pi$
$457$	−15324.5	−1.56860	−0.784302	+	0.620380i	$0.786978\pi$
$458$	6242.17	0.636850
$459$	0	0
$460$	−1805.72	−0.183026
$461$	7817.29	0.789777	0.394889	−	0.918729i	$-0.370783\pi$
$461$	7817.29	0.789777	0.394889	+	0.918729i	$0.370783\pi$
$462$	0	0
$463$	−15403.8	−1.54616	−0.773081	−	0.634307i	$-0.781285\pi$
$463$	−15403.8	−1.54616	−0.773081	+	0.634307i	$0.781285\pi$
$464$	−2498.44	−0.249973
$465$	0	0
$466$	−2720.07	−0.270396
$467$	−10624.9	−1.05281	−0.526406	−	0.850233i	$-0.676461\pi$
$467$	−10624.9	−1.05281	−0.526406	+	0.850233i	$0.676461\pi$
$468$	0	0
$469$	−27552.0	−2.71265
$470$	13137.8	1.28937
$471$	0	0
$472$	−9992.07	−0.974412
$473$	−2620.77	−0.254764
$474$	0	0
$475$	69.6817	0.00673098
$476$	2273.13	0.218884
$477$	0	0
$478$	18088.7	1.73088
$479$	−1566.23	−0.149401	−0.0747003	−	0.997206i	$-0.523800\pi$
$479$	−1566.23	−0.149401	−0.0747003	+	0.997206i	$0.523800\pi$
$480$	0	0
$481$	−17666.6	−1.67469
$482$	−12569.1	−1.18778
$483$	0	0
$484$	−149.440	−0.0140346
$485$	15399.0	1.44171
$486$	0	0
$487$	17547.2	1.63273	0.816365	−	0.577536i	$-0.195986\pi$
$487$	17547.2	1.63273	0.816365	+	0.577536i	$0.195986\pi$
$488$	12382.6	1.14863
$489$	0	0
$490$	11807.1	1.08855
$491$	−536.906	−0.0493488	−0.0246744	−	0.999696i	$-0.507855\pi$
$491$	−536.906	−0.0493488	−0.0246744	+	0.999696i	$0.507855\pi$
$492$	0	0
$493$	3181.75	0.290667
$494$	−3902.49	−0.355428
$495$	0	0
$496$	−14059.7	−1.27278
$497$	−11902.0	−1.07420
$498$	0	0
$499$	16209.3	1.45417	0.727083	−	0.686550i	$-0.240876\pi$
$499$	16209.3	1.45417	0.727083	+	0.686550i	$0.240876\pi$
$500$	−1750.41	−0.156561
$501$	0	0
$502$	2511.98	0.223337
$503$	−6827.35	−0.605202	−0.302601	−	0.953117i	$-0.597855\pi$
$503$	−6827.35	−0.605202	−0.302601	+	0.953117i	$0.597855\pi$
$504$	0	0
$505$	−6215.20	−0.547669
$506$	3797.56	0.333640
$507$	0	0
$508$	2102.23	0.183605
$509$	14453.1	1.25859	0.629295	−	0.777167i	$-0.283344\pi$
$509$	14453.1	1.25859	0.629295	+	0.777167i	$0.283344\pi$
$510$	0	0
$511$	−16323.3	−1.41312
$512$	13018.3	1.12370
$513$	0	0
$514$	−6629.72	−0.568919
$515$	7299.85	0.624602
$516$	0	0
$517$	5044.22	0.429100
$518$	15989.6	1.35626
$519$	0	0
$520$	20893.7	1.76202
$521$	−9338.88	−0.785305	−0.392652	−	0.919687i	$-0.628442\pi$
$521$	−9338.88	−0.785305	−0.392652	+	0.919687i	$0.628442\pi$
$522$	0	0
$523$	4041.70	0.337919	0.168959	−	0.985623i	$-0.445959\pi$
$523$	4041.70	0.337919	0.168959	+	0.985623i	$0.445959\pi$
$524$	1278.95	0.106624
$525$	0	0
$526$	−5223.17	−0.432968
$527$	17905.0	1.47999
$528$	0	0
$529$	5451.07	0.448021
$530$	−1531.34	−0.125504
$531$	0	0
$532$	−644.826	−0.0525503
$533$	33212.4	2.69904
$534$	0	0
$535$	−18790.3	−1.51846
$536$	24083.4	1.94075
$537$	0	0
$538$	17743.7	1.42191
$539$	4533.30	0.362269
$540$	0	0
$541$	908.291	0.0721820	0.0360910	−	0.999349i	$-0.488509\pi$
$541$	908.291	0.0721820	0.0360910	+	0.999349i	$0.488509\pi$
$542$	8921.67	0.707045
$543$	0	0
$544$	−3708.19	−0.292256
$545$	1242.93	0.0976904
$546$	0	0
$547$	8326.07	0.650818	0.325409	−	0.945573i	$-0.394498\pi$
$547$	8326.07	0.650818	0.325409	+	0.945573i	$0.394498\pi$
$548$	−1824.05	−0.142189
$549$	0	0
$550$	104.928	0.00813479
$551$	−902.577	−0.0697841
$552$	0	0
$553$	−21769.5	−1.67402
$554$	−7044.47	−0.540237
$555$	0	0
$556$	−1151.66	−0.0878442
$557$	−10458.8	−0.795607	−0.397803	−	0.917471i	$-0.630227\pi$
$557$	−10458.8	−0.795607	−0.397803	+	0.917471i	$0.630227\pi$
$558$	0	0
$559$	18814.5	1.42356
$560$	−15919.7	−1.20130
$561$	0	0
$562$	6608.72	0.496036
$563$	21585.1	1.61582	0.807909	−	0.589308i	$-0.200599\pi$
$563$	21585.1	1.61582	0.807909	+	0.589308i	$0.200599\pi$
$564$	0	0
$565$	−14059.2	−1.04686
$566$	−5473.16	−0.406456
$567$	0	0
$568$	10403.6	0.768530
$569$	17356.0	1.27874	0.639368	−	0.768901i	$-0.279196\pi$
$569$	17356.0	1.27874	0.639368	+	0.768901i	$0.279196\pi$
$570$	0	0
$571$	−400.224	−0.0293325	−0.0146662	−	0.999892i	$-0.504669\pi$
$571$	−400.224	−0.0293325	−0.0146662	+	0.999892i	$0.504669\pi$
$572$	1072.83	0.0784217
$573$	0	0
$574$	−30059.7	−2.18583
$575$	486.793	0.0353055
$576$	0	0
$577$	14222.6	1.02616	0.513079	−	0.858341i	$-0.328505\pi$
$577$	14222.6	1.02616	0.513079	+	0.858341i	$0.328505\pi$
$578$	1110.31	0.0799008
$579$	0	0
$580$	646.250	0.0462656
$581$	−4867.54	−0.347572
$582$	0	0
$583$	−587.952	−0.0417675
$584$	14268.3	1.01101
$585$	0	0
$586$	−19049.9	−1.34291
$587$	4349.55	0.305835	0.152918	−	0.988239i	$-0.451133\pi$
$587$	4349.55	0.305835	0.152918	+	0.988239i	$0.451133\pi$
$588$	0	0
$589$	−5079.16	−0.355320
$590$	−11918.1	−0.831625
$591$	0	0
$592$	−11766.2	−0.816871
$593$	−1775.25	−0.122936	−0.0614679	−	0.998109i	$-0.519578\pi$
$593$	−1775.25	−0.122936	−0.0614679	+	0.998109i	$0.519578\pi$
$594$	0	0
$595$	20273.6	1.39687
$596$	3810.96	0.261918
$597$	0	0
$598$	−27262.6	−1.86430
$599$	−13578.2	−0.926191	−0.463095	−	0.886308i	$-0.653261\pi$
$599$	−13578.2	−0.926191	−0.463095	+	0.886308i	$0.653261\pi$
$600$	0	0
$601$	−2298.00	−0.155969	−0.0779846	−	0.996955i	$-0.524848\pi$
$601$	−2298.00	−0.155969	−0.0779846	+	0.996955i	$0.524848\pi$
$602$	−17028.5	−1.15287
$603$	0	0
$604$	2630.06	0.177178
$605$	−1332.83	−0.0895655
$606$	0	0
$607$	22810.4	1.52528	0.762641	−	0.646822i	$-0.223902\pi$
$607$	22810.4	1.52528	0.762641	+	0.646822i	$0.223902\pi$
$608$	1051.91	0.0701657
$609$	0	0
$610$	14769.3	0.980315
$611$	−36212.4	−2.39771
$612$	0	0
$613$	−18023.6	−1.18755	−0.593773	−	0.804633i	$-0.702362\pi$
$613$	−18023.6	−1.18755	−0.593773	+	0.804633i	$0.702362\pi$
$614$	−3130.01	−0.205728
$615$	0	0
$616$	−7260.58	−0.474898
$617$	21288.3	1.38903	0.694516	−	0.719477i	$-0.255618\pi$
$617$	21288.3	1.38903	0.694516	+	0.719477i	$0.255618\pi$
$618$	0	0
$619$	−4587.09	−0.297853	−0.148926	−	0.988848i	$-0.547582\pi$
$619$	−4587.09	−0.297853	−0.148926	+	0.988848i	$0.547582\pi$
$620$	3636.71	0.235571
$621$	0	0
$622$	2915.14	0.187920
$623$	16660.2	1.07139
$624$	0	0
$625$	−15153.1	−0.969800
$626$	13113.2	0.837237
$627$	0	0
$628$	−877.230	−0.0557409
$629$	14984.2	0.949853
$630$	0	0
$631$	−2745.79	−0.173230	−0.0866151	−	0.996242i	$-0.527605\pi$
$631$	−2745.79	−0.173230	−0.0866151	+	0.996242i	$0.527605\pi$
$632$	19028.9	1.19767
$633$	0	0
$634$	5184.35	0.324758
$635$	18749.4	1.17173
$636$	0	0
$637$	−32544.5	−2.02427
$638$	−1359.11	−0.0843383
$639$	0	0
$640$	11301.4	0.698008
$641$	−12072.4	−0.743887	−0.371944	−	0.928255i	$-0.621308\pi$
$641$	−12072.4	−0.743887	−0.371944	+	0.928255i	$0.621308\pi$
$642$	0	0
$643$	16883.0	1.03546	0.517730	−	0.855544i	$-0.326777\pi$
$643$	16883.0	1.03546	0.517730	+	0.855544i	$0.326777\pi$
$644$	−4504.72	−0.275638
$645$	0	0
$646$	3309.95	0.201592
$647$	−32674.5	−1.98542	−0.992709	−	0.120535i	$-0.961539\pi$
$647$	−32674.5	−1.98542	−0.992709	+	0.120535i	$0.961539\pi$
$648$	0	0
$649$	−4575.90	−0.276764
$650$	−753.276	−0.0454552
$651$	0	0
$652$	104.310	0.00626548
$653$	29481.3	1.76676	0.883379	−	0.468659i	$-0.155263\pi$
$653$	29481.3	1.76676	0.883379	+	0.468659i	$0.155263\pi$
$654$	0	0
$655$	11406.7	0.680452
$656$	22119.9	1.31652
$657$	0	0
$658$	32774.9	1.94179
$659$	6240.30	0.368874	0.184437	−	0.982844i	$-0.440954\pi$
$659$	6240.30	0.368874	0.184437	+	0.982844i	$0.440954\pi$
$660$	0	0
$661$	−18494.9	−1.08830	−0.544151	−	0.838987i	$-0.683148\pi$
$661$	−18494.9	−1.08830	−0.544151	+	0.838987i	$0.683148\pi$
$662$	−20813.7	−1.22198
$663$	0	0
$664$	4254.74	0.248669
$665$	−5751.08	−0.335365
$666$	0	0
$667$	−6305.36	−0.366034
$668$	−1522.28	−0.0881721
$669$	0	0
$670$	28725.4	1.65636
$671$	5670.64	0.326248
$672$	0	0
$673$	−7613.95	−0.436101	−0.218051	−	0.975937i	$-0.569970\pi$
$673$	−7613.95	−0.436101	−0.218051	+	0.975937i	$0.569970\pi$
$674$	−7294.73	−0.416888
$675$	0	0
$676$	−4988.43	−0.283821
$677$	5839.81	0.331525	0.165762	−	0.986166i	$-0.446992\pi$
$677$	5839.81	0.331525	0.165762	+	0.986166i	$0.446992\pi$
$678$	0	0
$679$	38415.8	2.17123
$680$	−17721.3	−0.999383
$681$	0	0
$682$	−7648.28	−0.429425
$683$	11114.2	0.622654	0.311327	−	0.950303i	$-0.399227\pi$
$683$	11114.2	0.622654	0.311327	+	0.950303i	$0.399227\pi$
$684$	0	0
$685$	−16268.4	−0.907421
$686$	4940.03	0.274944
$687$	0	0
$688$	12530.7	0.694373
$689$	4220.90	0.233387
$690$	0	0
$691$	30466.0	1.67725	0.838626	−	0.544708i	$-0.183360\pi$
$691$	30466.0	1.67725	0.838626	+	0.544708i	$0.183360\pi$
$692$	−215.293	−0.0118269
$693$	0	0
$694$	29689.2	1.62390
$695$	−10271.5	−0.560603
$696$	0	0
$697$	−28169.5	−1.53084
$698$	19620.4	1.06396
$699$	0	0
$700$	−124.467	−0.00672058
$701$	7672.47	0.413388	0.206694	−	0.978406i	$-0.433730\pi$
$701$	7672.47	0.413388	0.206694	+	0.978406i	$0.433730\pi$
$702$	0	0
$703$	−4250.60	−0.228044
$704$	6212.29	0.332577
$705$	0	0
$706$	25527.0	1.36080
$707$	−15505.1	−0.824793
$708$	0	0
$709$	30321.3	1.60612	0.803062	−	0.595895i	$-0.203203\pi$
$709$	30321.3	1.60612	0.803062	+	0.595895i	$0.203203\pi$
$710$	12408.9	0.655912
$711$	0	0
$712$	−14562.8	−0.766523
$713$	−35482.8	−1.86373
$714$	0	0
$715$	9568.36	0.500470
$716$	−910.978	−0.0475487
$717$	0	0
$718$	−22741.6	−1.18205
$719$	−15462.1	−0.802001	−0.401001	−	0.916078i	$-0.631338\pi$
$719$	−15462.1	−0.802001	−0.401001	+	0.916078i	$0.631338\pi$
$720$	0	0
$721$	18210.9	0.940654
$722$	−938.944	−0.0483987
$723$	0	0
$724$	1942.58	0.0997173
$725$	−174.219	−0.00892460
$726$	0	0
$727$	12230.2	0.623924	0.311962	−	0.950095i	$-0.399014\pi$
$727$	12230.2	0.623924	0.311962	+	0.950095i	$0.399014\pi$
$728$	52123.6	2.65361
$729$	0	0
$730$	17018.6	0.862857
$731$	−15957.7	−0.807412
$732$	0	0
$733$	−3687.81	−0.185828	−0.0929142	−	0.995674i	$-0.529618\pi$
$733$	−3687.81	−0.185828	−0.0929142	+	0.995674i	$0.529618\pi$
$734$	−24546.3	−1.23436
$735$	0	0
$736$	7348.62	0.368035
$737$	11029.1	0.551235
$738$	0	0
$739$	10915.4	0.543340	0.271670	−	0.962391i	$-0.412424\pi$
$739$	10915.4	0.543340	0.271670	+	0.962391i	$0.412424\pi$
$740$	3043.46	0.151189
$741$	0	0
$742$	−3820.22	−0.189009
$743$	−12938.8	−0.638870	−0.319435	−	0.947608i	$-0.603493\pi$
$743$	−12938.8	−0.638870	−0.319435	+	0.947608i	$0.603493\pi$
$744$	0	0
$745$	33989.2	1.67150
$746$	−23015.1	−1.12955
$747$	0	0
$748$	−909.933	−0.0444792
$749$	−46876.2	−2.28681
$750$	0	0
$751$	30617.1	1.48766	0.743830	−	0.668368i	$-0.233007\pi$
$751$	30617.1	1.48766	0.743830	+	0.668368i	$0.233007\pi$
$752$	−24118.0	−1.16954
$753$	0	0
$754$	9757.06	0.471262
$755$	23457.0	1.13071
$756$	0	0
$757$	−5464.29	−0.262356	−0.131178	−	0.991359i	$-0.541876\pi$
$757$	−5464.29	−0.262356	−0.131178	+	0.991359i	$0.541876\pi$
$758$	−16409.1	−0.786285
$759$	0	0
$760$	5027.06	0.239935
$761$	−2607.44	−0.124204	−0.0621022	−	0.998070i	$-0.519780\pi$
$761$	−2607.44	−0.124204	−0.0621022	+	0.998070i	$0.519780\pi$
$762$	0	0
$763$	3100.74	0.147122
$764$	2523.33	0.119491
$765$	0	0
$766$	22726.6	1.07199
$767$	32850.4	1.54649
$768$	0	0
$769$	32315.3	1.51537	0.757684	−	0.652621i	$-0.226331\pi$
$769$	32315.3	1.51537	0.757684	+	0.652621i	$0.226331\pi$
$770$	−8660.07	−0.405308
$771$	0	0
$772$	−4424.64	−0.206278
$773$	−2930.47	−0.136354	−0.0681771	−	0.997673i	$-0.521718\pi$
$773$	−2930.47	−0.136354	−0.0681771	+	0.997673i	$0.521718\pi$
$774$	0	0
$775$	−980.401	−0.0454413
$776$	−33579.5	−1.55339
$777$	0	0
$778$	8629.85	0.397680
$779$	7990.94	0.367529
$780$	0	0
$781$	4764.36	0.218287
$782$	23123.2	1.05739
$783$	0	0
$784$	−21675.1	−0.987384
$785$	−7823.85	−0.355726
$786$	0	0
$787$	1978.09	0.0895949	0.0447974	−	0.998996i	$-0.485736\pi$
$787$	1978.09	0.0895949	0.0447974	+	0.998996i	$0.485736\pi$
$788$	5383.38	0.243369
$789$	0	0
$790$	22696.7	1.02217
$791$	−35073.6	−1.57658
$792$	0	0
$793$	−40709.5	−1.82300
$794$	13656.6	0.610394
$795$	0	0
$796$	−6620.92	−0.294815
$797$	−9527.89	−0.423457	−0.211728	−	0.977329i	$-0.567909\pi$
$797$	−9527.89	−0.423457	−0.211728	+	0.977329i	$0.567909\pi$
$798$	0	0
$799$	30714.0	1.35993
$800$	203.045	0.00897339
$801$	0	0
$802$	8108.63	0.357015
$803$	6534.23	0.287158
$804$	0	0
$805$	−40176.8	−1.75906
$806$	54906.9	2.39952
$807$	0	0
$808$	13553.1	0.590094
$809$	34176.1	1.48525	0.742626	−	0.669706i	$-0.233580\pi$
$809$	34176.1	1.48525	0.742626	+	0.669706i	$0.233580\pi$
$810$	0	0
$811$	−35496.9	−1.53695	−0.768474	−	0.639881i	$-0.778984\pi$
$811$	−35496.9	−1.53695	−0.768474	+	0.639881i	$0.778984\pi$
$812$	1612.20	0.0696763
$813$	0	0
$814$	−6400.63	−0.275604
$815$	930.321	0.0399850
$816$	0	0
$817$	4526.78	0.193846
$818$	7540.43	0.322304
$819$	0	0
$820$	−5721.56	−0.243665
$821$	8003.37	0.340219	0.170109	−	0.985425i	$-0.445588\pi$
$821$	8003.37	0.340219	0.170109	+	0.985425i	$0.445588\pi$
$822$	0	0
$823$	−20192.9	−0.855262	−0.427631	−	0.903953i	$-0.640652\pi$
$823$	−20192.9	−0.855262	−0.427631	+	0.903953i	$0.640652\pi$
$824$	−15918.3	−0.672986
$825$	0	0
$826$	−29732.0	−1.25243
$827$	3898.17	0.163909	0.0819544	−	0.996636i	$-0.473884\pi$
$827$	3898.17	0.163909	0.0819544	+	0.996636i	$0.473884\pi$
$828$	0	0
$829$	11199.5	0.469210	0.234605	−	0.972091i	$-0.424620\pi$
$829$	11199.5	0.469210	0.234605	+	0.972091i	$0.424620\pi$
$830$	5074.85	0.212230
$831$	0	0
$832$	−44598.0	−1.85836
$833$	27603.0	1.14812
$834$	0	0
$835$	−13577.0	−0.562695
$836$	258.124	0.0106787
$837$	0	0
$838$	16949.1	0.698686
$839$	19437.5	0.799828	0.399914	−	0.916553i	$-0.369040\pi$
$839$	19437.5	0.799828	0.399914	+	0.916553i	$0.369040\pi$
$840$	0	0
$841$	−22132.4	−0.907473
$842$	32164.6	1.31647
$843$	0	0
$844$	5132.71	0.209331
$845$	−44490.9	−1.81128
$846$	0	0
$847$	−3325.01	−0.134886
$848$	2811.18	0.113840
$849$	0	0
$850$	638.900	0.0257813
$851$	−29694.5	−1.19614
$852$	0	0
$853$	−25303.5	−1.01568	−0.507841	−	0.861451i	$-0.669556\pi$
$853$	−25303.5	−1.01568	−0.507841	+	0.861451i	$0.669556\pi$
$854$	36845.1	1.47636
$855$	0	0
$856$	40974.7	1.63608
$857$	48720.5	1.94196	0.970980	−	0.239162i	$-0.0768727\pi$
$857$	48720.5	1.94196	0.970980	+	0.239162i	$0.0768727\pi$
$858$	0	0
$859$	13480.2	0.535437	0.267718	−	0.963497i	$-0.413730\pi$
$859$	13480.2	0.535437	0.267718	+	0.963497i	$0.413730\pi$
$860$	−3241.20	−0.128516
$861$	0	0
$862$	45888.8	1.81320
$863$	−5021.66	−0.198076	−0.0990378	−	0.995084i	$-0.531576\pi$
$863$	−5021.66	−0.198076	−0.0990378	+	0.995084i	$0.531576\pi$
$864$	0	0
$865$	−1920.16	−0.0754768
$866$	−38089.9	−1.49463
$867$	0	0
$868$	9072.50	0.354771
$869$	8714.34	0.340177
$870$	0	0
$871$	−79177.5	−3.08017
$872$	−2710.37	−0.105258
$873$	0	0
$874$	−6559.42	−0.253862
$875$	−38946.2	−1.50471
$876$	0	0
$877$	−28513.7	−1.09788	−0.548939	−	0.835863i	$-0.684968\pi$
$877$	−28513.7	−1.09788	−0.548939	+	0.835863i	$0.684968\pi$
$878$	43339.8	1.66588
$879$	0	0
$880$	6372.65	0.244116
$881$	12611.5	0.482285	0.241142	−	0.970490i	$-0.422478\pi$
$881$	12611.5	0.482285	0.241142	+	0.970490i	$0.422478\pi$
$882$	0	0
$883$	22015.0	0.839029	0.419515	−	0.907749i	$-0.362200\pi$
$883$	22015.0	0.839029	0.419515	+	0.907749i	$0.362200\pi$
$884$	6532.40	0.248539
$885$	0	0
$886$	29518.0	1.11928
$887$	27627.5	1.04582	0.522909	−	0.852389i	$-0.324847\pi$
$887$	27627.5	1.04582	0.522909	+	0.852389i	$0.324847\pi$
$888$	0	0
$889$	46774.1	1.76463
$890$	−17369.8	−0.654199
$891$	0	0
$892$	771.329	0.0289529
$893$	−8712.75	−0.326496
$894$	0	0
$895$	−8124.85	−0.303445
$896$	28193.5	1.05120
$897$	0	0
$898$	−3191.39	−0.118595
$899$	12699.0	0.471118
$900$	0	0
$901$	−3580.01	−0.132372
$902$	12032.9	0.444180
$903$	0	0
$904$	30658.0	1.12795
$905$	17325.5	0.636375
$906$	0	0
$907$	−6019.67	−0.220375	−0.110187	−	0.993911i	$-0.535145\pi$
$907$	−6019.67	−0.220375	−0.110187	+	0.993911i	$0.535145\pi$
$908$	−4245.54	−0.155169
$909$	0	0
$910$	62170.5	2.26476
$911$	21231.9	0.772168	0.386084	−	0.922464i	$-0.373827\pi$
$911$	21231.9	0.772168	0.386084	+	0.922464i	$0.373827\pi$
$912$	0	0
$913$	1948.47	0.0706299
$914$	39858.4	1.44245
$915$	0	0
$916$	2964.04	0.106916
$917$	28456.3	1.02476
$918$	0	0
$919$	1083.33	0.0388857	0.0194428	−	0.999811i	$-0.493811\pi$
$919$	1083.33	0.0388857	0.0194428	+	0.999811i	$0.493811\pi$
$920$	35118.8	1.25851
$921$	0	0
$922$	−20332.4	−0.726260
$923$	−34203.3	−1.21973
$924$	0	0
$925$	−820.469	−0.0291642
$926$	40064.4	1.42181
$927$	0	0
$928$	−2630.01	−0.0930326
$929$	−26411.6	−0.932761	−0.466381	−	0.884584i	$-0.654442\pi$
$929$	−26411.6	−0.932761	−0.466381	+	0.884584i	$0.654442\pi$
$930$	0	0
$931$	−7830.24	−0.275645
$932$	−1291.60	−0.0453946
$933$	0	0
$934$	27635.0	0.968140
$935$	−8115.52	−0.283857
$936$	0	0
$937$	−13033.5	−0.454415	−0.227207	−	0.973846i	$-0.572960\pi$
$937$	−13033.5	−0.454415	−0.227207	+	0.973846i	$0.572960\pi$
$938$	71661.4	2.49449
$939$	0	0
$940$	6238.38	0.216461
$941$	−10411.9	−0.360700	−0.180350	−	0.983603i	$-0.557723\pi$
$941$	−10411.9	−0.360700	−0.180350	+	0.983603i	$0.557723\pi$
$942$	0	0
$943$	55824.3	1.92777
$944$	21878.8	0.754337
$945$	0	0
$946$	6816.50	0.234274
$947$	39294.0	1.34835	0.674173	−	0.738573i	$-0.264500\pi$
$947$	39294.0	1.34835	0.674173	+	0.738573i	$0.264500\pi$
$948$	0	0
$949$	−46909.2	−1.60457
$950$	−181.239	−0.00618965
$951$	0	0
$952$	−44209.3	−1.50508
$953$	−14222.6	−0.483438	−0.241719	−	0.970346i	$-0.577711\pi$
$953$	−14222.6	−0.483438	−0.241719	+	0.970346i	$0.577711\pi$
$954$	0	0
$955$	22505.1	0.762564
$956$	8589.27	0.290583
$957$	0	0
$958$	4073.69	0.137385
$959$	−40584.8	−1.36658
$960$	0	0
$961$	41671.3	1.39879
$962$	45950.0	1.54001
$963$	0	0
$964$	−5968.34	−0.199406
$965$	−39462.6	−1.31642
$966$	0	0
$967$	−2765.09	−0.0919538	−0.0459769	−	0.998943i	$-0.514640\pi$
$967$	−2765.09	−0.0919538	−0.0459769	+	0.998943i	$0.514640\pi$
$968$	2906.41	0.0965037
$969$	0	0
$970$	−40052.0	−1.32576
$971$	38472.4	1.27151	0.635757	−	0.771890i	$-0.280688\pi$
$971$	38472.4	1.27151	0.635757	+	0.771890i	$0.280688\pi$
$972$	0	0
$973$	−25624.3	−0.844271
$974$	−45639.5	−1.50142
$975$	0	0
$976$	−27113.0	−0.889209
$977$	19385.6	0.634802	0.317401	−	0.948291i	$-0.397190\pi$
$977$	19385.6	0.634802	0.317401	+	0.948291i	$0.397190\pi$
$978$	0	0
$979$	−6669.08	−0.217717
$980$	5606.49	0.182748
$981$	0	0
$982$	1396.47	0.0453799
$983$	−3670.52	−0.119096	−0.0595480	−	0.998225i	$-0.518966\pi$
$983$	−3670.52	−0.119096	−0.0595480	+	0.998225i	$0.518966\pi$
$984$	0	0
$985$	48013.4	1.55313
$986$	−8275.58	−0.267290
$987$	0	0
$988$	−1853.07	−0.0596699
$989$	31623.9	1.01677
$990$	0	0
$991$	−55604.3	−1.78237	−0.891186	−	0.453639i	$-0.850126\pi$
$991$	−55604.3	−1.78237	−0.891186	+	0.453639i	$0.850126\pi$
$992$	−14800.1	−0.473693
$993$	0	0
$994$	30956.5	0.987808
$995$	−59050.8	−1.88144
$996$	0	0
$997$	13889.9	0.441220	0.220610	−	0.975362i	$-0.429195\pi$
$997$	13889.9	0.441220	0.220610	+	0.975362i	$0.429195\pi$
$998$	−42159.7	−1.33722
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.4.a.n.1.8	✓	23
3.2	odd	2		1881.4.a.o.1.16	yes	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1881.4.a.n.1.8	✓	23	1.1	even	1	trivial
1881.4.a.o.1.16	yes	23	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q-2.60095 q^{2} -1.23504 q^{4} -11.0151 q^{5} -27.4794 q^{7} +24.0199 q^{8} +28.6498 q^{10} +11.0000 q^{11} -78.9689 q^{13} +71.4727 q^{14} -52.5943 q^{16} +66.9785 q^{17} -19.0000 q^{19} +13.6041 q^{20} -28.6105 q^{22} -132.733 q^{23} -3.66746 q^{25} +205.394 q^{26} +33.9382 q^{28} +47.5040 q^{29} +267.324 q^{31} -55.3639 q^{32} -174.208 q^{34} +302.689 q^{35} +223.716 q^{37} +49.4181 q^{38} -264.582 q^{40} -420.576 q^{41} -238.252 q^{43} -13.5855 q^{44} +345.233 q^{46} +458.566 q^{47} +412.118 q^{49} +9.53889 q^{50} +97.5298 q^{52} -53.4502 q^{53} -121.166 q^{55} -660.053 q^{56} -123.556 q^{58} -415.991 q^{59} +515.513 q^{61} -695.298 q^{62} +564.754 q^{64} +869.851 q^{65} +1002.64 q^{67} -82.7212 q^{68} -787.279 q^{70} +433.124 q^{71} +594.021 q^{73} -581.875 q^{74} +23.4658 q^{76} -302.273 q^{77} +792.213 q^{79} +579.332 q^{80} +1093.90 q^{82} +177.134 q^{83} -737.775 q^{85} +619.682 q^{86} +264.219 q^{88} -606.280 q^{89} +2170.02 q^{91} +163.931 q^{92} -1192.71 q^{94} +209.287 q^{95} -1397.99 q^{97} -1071.90 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 4 q^{2} + 96 q^{4} - 14 q^{7} - 48 q^{8} - 124 q^{10} + 253 q^{11} - 150 q^{13} - 152 q^{14} + 444 q^{16} + 68 q^{17} - 437 q^{19} - 80 q^{20} - 44 q^{22} - 414 q^{23} + 383 q^{25} - 464 q^{26} - 384 q^{28}+ \cdots - 7876 q^{98}+O(q^{100}) \) 23 * q - 4 * q^2 + 96 * q^4 - 14 * q^7 - 48 * q^8 - 124 * q^10 + 253 * q^11 - 150 * q^13 - 152 * q^14 + 444 * q^16 + 68 * q^17 - 437 * q^19 - 80 * q^20 - 44 * q^22 - 414 * q^23 + 383 * q^25 - 464 * q^26 - 384 * q^28 - 604 * q^29 + 136 * q^31 - 448 * q^32 - 114 * q^34 - 312 * q^35 - 134 * q^37 + 76 * q^38 - 1518 * q^40 - 540 * q^41 - 742 * q^43 + 1056 * q^44 + 230 * q^46 - 426 * q^47 + 1059 * q^49 - 1322 * q^50 - 320 * q^52 - 1164 * q^53 - 5058 * q^56 + 2572 * q^58 - 1490 * q^59 - 666 * q^61 - 5010 * q^62 + 5582 * q^64 - 2836 * q^65 + 324 * q^67 + 656 * q^68 + 3226 * q^70 - 4188 * q^71 - 242 * q^73 - 4730 * q^74 - 1824 * q^76 - 154 * q^77 - 436 * q^79 - 4914 * q^80 - 52 * q^82 - 1322 * q^83 - 3650 * q^85 - 3298 * q^86 - 528 * q^88 - 1630 * q^89 + 876 * q^91 - 6110 * q^92 + 2494 * q^94 - 5162 * q^97 - 7876 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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