LMFDB - Embedded newform 1815.4.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, 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("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.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:	$107.088466660$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 44x^{4} + 495x^{2} - 1200 $ x^6 - 44x^4 + 495x^2 - 1200
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.82790$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.82790 q^{2} +3.00000 q^{3} -4.65878 q^{4} +5.00000 q^{5} -5.48370 q^{6} +15.0916 q^{7} +23.1390 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.82790 q^{2} +3.00000 q^{3} -4.65878 q^{4} +5.00000 q^{5} -5.48370 q^{6} +15.0916 q^{7} +23.1390 q^{8} +9.00000 q^{9} -9.13951 q^{10} -13.9763 q^{12} +11.4358 q^{13} -27.5859 q^{14} +15.0000 q^{15} -5.02557 q^{16} -11.9707 q^{17} -16.4511 q^{18} +91.0843 q^{19} -23.2939 q^{20} +45.2747 q^{21} +103.172 q^{23} +69.4170 q^{24} +25.0000 q^{25} -20.9034 q^{26} +27.0000 q^{27} -70.3082 q^{28} +99.3326 q^{29} -27.4185 q^{30} -39.9053 q^{31} -175.926 q^{32} +21.8813 q^{34} +75.4578 q^{35} -41.9290 q^{36} +149.172 q^{37} -166.493 q^{38} +34.3073 q^{39} +115.695 q^{40} +135.891 q^{41} -82.7576 q^{42} -304.928 q^{43} +45.0000 q^{45} -188.588 q^{46} +113.073 q^{47} -15.0767 q^{48} -115.245 q^{49} -45.6975 q^{50} -35.9122 q^{51} -53.2766 q^{52} +552.888 q^{53} -49.3533 q^{54} +349.203 q^{56} +273.253 q^{57} -181.570 q^{58} -251.961 q^{59} -69.8817 q^{60} +211.282 q^{61} +72.9430 q^{62} +135.824 q^{63} +361.779 q^{64} +57.1788 q^{65} +947.582 q^{67} +55.7690 q^{68} +309.515 q^{69} -137.929 q^{70} -175.376 q^{71} +208.251 q^{72} -768.287 q^{73} -272.671 q^{74} +75.0000 q^{75} -424.342 q^{76} -62.7103 q^{78} +710.879 q^{79} -25.1279 q^{80} +81.0000 q^{81} -248.395 q^{82} -902.213 q^{83} -210.925 q^{84} -59.8537 q^{85} +557.377 q^{86} +297.998 q^{87} -482.884 q^{89} -82.2556 q^{90} +172.583 q^{91} -480.654 q^{92} -119.716 q^{93} -206.687 q^{94} +455.422 q^{95} -527.777 q^{96} +724.303 q^{97} +210.656 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10}) $ 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9	$ 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9} + 120 q^{12} + 116 q^{14} + 90 q^{15} + 164 q^{16} + 200 q^{20} + 56 q^{23} + 150 q^{25} + 292 q^{26} + 162 q^{27} + 576 q^{31} - 92 q^{34} + 360 q^{36} + 332 q^{37} + 496 q^{38} + 348 q^{42} + 270 q^{45} + 96 q^{47} + 492 q^{48} + 454 q^{49} + 308 q^{53} - 652 q^{56} + 1784 q^{58} + 2080 q^{59} + 600 q^{60} + 1928 q^{64} - 1168 q^{67} + 168 q^{69} + 580 q^{70} + 1064 q^{71} + 450 q^{75} + 876 q^{78} + 820 q^{80} + 486 q^{81} + 24 q^{82} + 5412 q^{86} - 684 q^{89} + 2744 q^{91} + 1368 q^{92} + 1728 q^{93} + 2812 q^{97}+O(q^{100}) $ 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9 + 120 * q^12 + 116 * q^14 + 90 * q^15 + 164 * q^16 + 200 * q^20 + 56 * q^23 + 150 * q^25 + 292 * q^26 + 162 * q^27 + 576 * q^31 - 92 * q^34 + 360 * q^36 + 332 * q^37 + 496 * q^38 + 348 * q^42 + 270 * q^45 + 96 * q^47 + 492 * q^48 + 454 * q^49 + 308 * q^53 - 652 * q^56 + 1784 * q^58 + 2080 * q^59 + 600 * q^60 + 1928 * q^64 - 1168 * q^67 + 168 * q^69 + 580 * q^70 + 1064 * q^71 + 450 * q^75 + 876 * q^78 + 820 * q^80 + 486 * q^81 + 24 * q^82 + 5412 * q^86 - 684 * q^89 + 2744 * q^91 + 1368 * q^92 + 1728 * q^93 + 2812 * q^97

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 $

$2$	−1.82790	−0.646261	−0.323130	−	0.946354i	$-0.604735\pi$
$2$	−1.82790	−0.646261	−0.323130	+	0.946354i	$0.604735\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.65878	−0.582347
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−5.48370	−0.373119
$7$	15.0916	0.814867	0.407434	−	0.913235i	$-0.366424\pi$
$7$	15.0916	0.814867	0.407434	+	0.913235i	$0.366424\pi$
$8$	23.1390	1.02261
$9$	9.00000	0.333333
$10$	−9.13951	−0.289017
$11$	0	0
$11$	0	0
$12$	−13.9763	−0.336218
$13$	11.4358	0.243977	0.121989	−	0.992531i	$-0.461073\pi$
$13$	11.4358	0.243977	0.121989	+	0.992531i	$0.461073\pi$
$14$	−27.5859	−0.526617
$15$	15.0000	0.258199
$16$	−5.02557	−0.0785246
$17$	−11.9707	−0.170784	−0.0853920	−	0.996347i	$-0.527214\pi$
$17$	−11.9707	−0.170784	−0.0853920	+	0.996347i	$0.527214\pi$
$18$	−16.4511	−0.215420
$19$	91.0843	1.09980	0.549899	−	0.835231i	$-0.314666\pi$
$19$	91.0843	1.09980	0.549899	+	0.835231i	$0.314666\pi$
$20$	−23.2939	−0.260434
$21$	45.2747	0.470464
$22$	0	0
$23$	103.172	0.935339	0.467669	−	0.883903i	$-0.345094\pi$
$23$	103.172	0.935339	0.467669	+	0.883903i	$0.345094\pi$
$24$	69.4170	0.590403
$25$	25.0000	0.200000
$26$	−20.9034	−0.157673
$27$	27.0000	0.192450
$28$	−70.3082	−0.474536
$29$	99.3326	0.636055	0.318028	−	0.948081i	$-0.396980\pi$
$29$	99.3326	0.636055	0.318028	+	0.948081i	$0.396980\pi$
$30$	−27.4185	−0.166864
$31$	−39.9053	−0.231200	−0.115600	−	0.993296i	$-0.536879\pi$
$31$	−39.9053	−0.231200	−0.115600	+	0.993296i	$0.536879\pi$
$32$	−175.926	−0.971861
$33$	0	0
$34$	21.8813	0.110371
$35$	75.4578	0.364420
$36$	−41.9290	−0.194116
$37$	149.172	0.662802	0.331401	−	0.943490i	$-0.392479\pi$
$37$	149.172	0.662802	0.331401	+	0.943490i	$0.392479\pi$
$38$	−166.493	−0.710757
$39$	34.3073	0.140860
$40$	115.695	0.457325
$41$	135.891	0.517623	0.258812	−	0.965928i	$-0.416669\pi$
$41$	135.891	0.517623	0.258812	+	0.965928i	$0.416669\pi$
$42$	−82.7576	−0.304042
$43$	−304.928	−1.08142	−0.540710	−	0.841209i	$-0.681844\pi$
$43$	−304.928	−1.08142	−0.540710	+	0.841209i	$0.681844\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	−188.588	−0.604473
$47$	113.073	0.350924	0.175462	−	0.984486i	$-0.443858\pi$
$47$	113.073	0.350924	0.175462	+	0.984486i	$0.443858\pi$
$48$	−15.0767	−0.0453362
$49$	−115.245	−0.335991
$50$	−45.6975	−0.129252
$51$	−35.9122	−0.0986022
$52$	−53.2766	−0.142080
$53$	552.888	1.43292	0.716462	−	0.697626i	$-0.245760\pi$
$53$	552.888	1.43292	0.716462	+	0.697626i	$0.245760\pi$
$54$	−49.3533	−0.124373
$55$	0	0
$56$	349.203	0.833290
$57$	273.253	0.634969
$58$	−181.570	−0.411057
$59$	−251.961	−0.555975	−0.277988	−	0.960585i	$-0.589667\pi$
$59$	−251.961	−0.555975	−0.277988	+	0.960585i	$0.589667\pi$
$60$	−69.8817	−0.150361
$61$	211.282	0.443473	0.221737	−	0.975107i	$-0.428828\pi$
$61$	211.282	0.443473	0.221737	+	0.975107i	$0.428828\pi$
$62$	72.9430	0.149416
$63$	135.824	0.271622
$64$	361.779	0.706600
$65$	57.1788	0.109110
$66$	0	0
$67$	947.582	1.72784	0.863922	−	0.503625i	$-0.168001\pi$
$67$	947.582	1.72784	0.863922	+	0.503625i	$0.168001\pi$
$68$	55.7690	0.0994556
$69$	309.515	0.540018
$70$	−137.929	−0.235510
$71$	−175.376	−0.293146	−0.146573	−	0.989200i	$-0.546824\pi$
$71$	−175.376	−0.293146	−0.146573	+	0.989200i	$0.546824\pi$
$72$	208.251	0.340870
$73$	−768.287	−1.23180	−0.615898	−	0.787826i	$-0.711207\pi$
$73$	−768.287	−1.23180	−0.615898	+	0.787826i	$0.711207\pi$
$74$	−272.671	−0.428343
$75$	75.0000	0.115470
$76$	−424.342	−0.640465
$77$	0	0
$78$	−62.7103	−0.0910326
$79$	710.879	1.01241	0.506204	−	0.862414i	$-0.331048\pi$
$79$	710.879	1.01241	0.506204	+	0.862414i	$0.331048\pi$
$80$	−25.1279	−0.0351173
$81$	81.0000	0.111111
$82$	−248.395	−0.334519
$83$	−902.213	−1.19314	−0.596571	−	0.802561i	$-0.703470\pi$
$83$	−902.213	−1.19314	−0.596571	+	0.802561i	$0.703470\pi$
$84$	−210.925	−0.273973
$85$	−59.8537	−0.0763770
$86$	557.377	0.698879
$87$	297.998	0.367227
$88$	0	0
$89$	−482.884	−0.575119	−0.287560	−	0.957763i	$-0.592844\pi$
$89$	−482.884	−0.575119	−0.287560	+	0.957763i	$0.592844\pi$
$90$	−82.2556	−0.0963388
$91$	172.583	0.198809
$92$	−480.654	−0.544692
$93$	−119.716	−0.133484
$94$	−206.687	−0.226788
$95$	455.422	0.491845
$96$	−527.777	−0.561104
$97$	724.303	0.758163	0.379082	−	0.925363i	$-0.376240\pi$
$97$	724.303	0.758163	0.379082	+	0.925363i	$0.376240\pi$
$98$	210.656	0.217138
$99$	0	0
$100$	−116.469	−0.116469
$101$	−73.5266	−0.0724374	−0.0362187	−	0.999344i	$-0.511531\pi$
$101$	−73.5266	−0.0724374	−0.0362187	+	0.999344i	$0.511531\pi$
$102$	65.6439	0.0637227
$103$	−156.047	−0.149279	−0.0746395	−	0.997211i	$-0.523781\pi$
$103$	−156.047	−0.149279	−0.0746395	+	0.997211i	$0.523781\pi$
$104$	264.612	0.249493
$105$	226.373	0.210398
$106$	−1010.62	−0.926042
$107$	−1998.62	−1.80573	−0.902867	−	0.429920i	$-0.858542\pi$
$107$	−1998.62	−1.80573	−0.902867	+	0.429920i	$0.858542\pi$
$108$	−125.787	−0.112073
$109$	−642.090	−0.564230	−0.282115	−	0.959381i	$-0.591036\pi$
$109$	−642.090	−0.564230	−0.282115	+	0.959381i	$0.591036\pi$
$110$	0	0
$111$	447.515	0.382669
$112$	−75.8437	−0.0639871
$113$	227.521	0.189411	0.0947054	−	0.995505i	$-0.469809\pi$
$113$	227.521	0.189411	0.0947054	+	0.995505i	$0.469809\pi$
$114$	−499.479	−0.410355
$115$	515.859	0.418296
$116$	−462.768	−0.370405
$117$	102.922	0.0813258
$118$	460.560	0.359305
$119$	−180.657	−0.139166
$120$	347.085	0.264036
$121$	0	0
$122$	−386.202	−0.286599
$123$	407.672	0.298850
$124$	185.910	0.134639
$125$	125.000	0.0894427
$126$	−248.273	−0.175539
$127$	1189.73	0.831274	0.415637	−	0.909531i	$-0.363559\pi$
$127$	1189.73	0.831274	0.415637	+	0.909531i	$0.363559\pi$
$128$	746.109	0.515213
$129$	−914.783	−0.624358
$130$	−104.517	−0.0705135
$131$	−1401.48	−0.934713	−0.467357	−	0.884069i	$-0.654794\pi$
$131$	−1401.48	−0.934713	−0.467357	+	0.884069i	$0.654794\pi$
$132$	0	0
$133$	1374.60	0.896190
$134$	−1732.09	−1.11664
$135$	135.000	0.0860663
$136$	−276.991	−0.174645
$137$	826.015	0.515119	0.257559	−	0.966262i	$-0.417082\pi$
$137$	826.015	0.515119	0.257559	+	0.966262i	$0.417082\pi$
$138$	−565.763	−0.348992
$139$	−1920.31	−1.17179	−0.585895	−	0.810387i	$-0.699257\pi$
$139$	−1920.31	−1.17179	−0.585895	+	0.810387i	$0.699257\pi$
$140$	−351.541	−0.212219
$141$	339.220	0.202606
$142$	320.571	0.189448
$143$	0	0
$144$	−45.2302	−0.0261749
$145$	496.663	0.284453
$146$	1404.35	0.796062
$147$	−345.735	−0.193985
$148$	−694.958	−0.385981
$149$	−1057.30	−0.581325	−0.290662	−	0.956826i	$-0.593876\pi$
$149$	−1057.30	−0.581325	−0.290662	+	0.956826i	$0.593876\pi$
$150$	−137.093	−0.0746238
$151$	−2243.75	−1.20923	−0.604614	−	0.796519i	$-0.706673\pi$
$151$	−2243.75	−1.20923	−0.604614	+	0.796519i	$0.706673\pi$
$152$	2107.60	1.12466
$153$	−107.737	−0.0569280
$154$	0	0
$155$	−199.527	−0.103396
$156$	−159.830	−0.0820297
$157$	382.672	0.194526	0.0972628	−	0.995259i	$-0.468991\pi$
$157$	382.672	0.194526	0.0972628	+	0.995259i	$0.468991\pi$
$158$	−1299.42	−0.654279
$159$	1658.66	0.827299
$160$	−879.628	−0.434630
$161$	1557.02	0.762177
$162$	−148.060	−0.0718067
$163$	2513.97	1.20803	0.604017	−	0.796971i	$-0.293566\pi$
$163$	2513.97	1.20803	0.604017	+	0.796971i	$0.293566\pi$
$164$	−633.084	−0.301436
$165$	0	0
$166$	1649.16	0.771080
$167$	4280.74	1.98356	0.991778	−	0.127971i	$-0.0408464\pi$
$167$	4280.74	1.98356	0.991778	+	0.127971i	$0.0408464\pi$
$168$	1047.61	0.481100
$169$	−2066.22	−0.940475
$170$	109.407	0.0493594
$171$	819.759	0.366600
$172$	1420.59	0.629761
$173$	2214.27	0.973110	0.486555	−	0.873650i	$-0.338253\pi$
$173$	2214.27	0.973110	0.486555	+	0.873650i	$0.338253\pi$
$174$	−544.711	−0.237324
$175$	377.289	0.162973
$176$	0	0
$177$	−755.883	−0.320992
$178$	882.664	0.371677
$179$	1570.64	0.655837	0.327918	−	0.944706i	$-0.393653\pi$
$179$	1570.64	0.655837	0.327918	+	0.944706i	$0.393653\pi$
$180$	−209.645	−0.0868112
$181$	2867.18	1.17743	0.588717	−	0.808339i	$-0.299633\pi$
$181$	2867.18	1.17743	0.588717	+	0.808339i	$0.299633\pi$
$182$	−315.465	−0.128483
$183$	633.845	0.256039
$184$	2387.29	0.956486
$185$	745.859	0.296414
$186$	218.829	0.0862652
$187$	0	0
$188$	−526.783	−0.204360
$189$	407.472	0.156821
$190$	−832.466	−0.317860
$191$	−2081.99	−0.788732	−0.394366	−	0.918953i	$-0.629036\pi$
$191$	−2081.99	−0.788732	−0.394366	+	0.918953i	$0.629036\pi$
$192$	1085.34	0.407956
$193$	−2362.68	−0.881190	−0.440595	−	0.897706i	$-0.645232\pi$
$193$	−2362.68	−0.881190	−0.440595	+	0.897706i	$0.645232\pi$
$194$	−1323.95	−0.489971
$195$	171.536	0.0629947
$196$	536.901	0.195664
$197$	232.619	0.0841292	0.0420646	−	0.999115i	$-0.486606\pi$
$197$	232.619	0.0841292	0.0420646	+	0.999115i	$0.486606\pi$
$198$	0	0
$199$	563.211	0.200628	0.100314	−	0.994956i	$-0.468015\pi$
$199$	563.211	0.200628	0.100314	+	0.994956i	$0.468015\pi$
$200$	578.475	0.204522
$201$	2842.75	0.997572
$202$	134.399	0.0468134
$203$	1499.08	0.518301
$204$	167.307	0.0574207
$205$	679.453	0.231488
$206$	285.238	0.0964731
$207$	928.546	0.311780
$208$	−57.4712	−0.0191582
$209$	0	0
$210$	−413.788	−0.135972
$211$	48.3870	0.0157872	0.00789360	−	0.999969i	$-0.497487\pi$
$211$	48.3870	0.0157872	0.00789360	+	0.999969i	$0.497487\pi$
$212$	−2575.78	−0.834459
$213$	−526.129	−0.169248
$214$	3653.27	1.16697
$215$	−1524.64	−0.483625
$216$	624.753	0.196801
$217$	−602.233	−0.188398
$218$	1173.68	0.364639
$219$	−2304.86	−0.711178
$220$	0	0
$221$	−136.894	−0.0416675
$222$	−818.014	−0.247304
$223$	2363.27	0.709669	0.354834	−	0.934929i	$-0.384537\pi$
$223$	2363.27	0.709669	0.354834	+	0.934929i	$0.384537\pi$
$224$	−2654.99	−0.791938
$225$	225.000	0.0666667
$226$	−415.887	−0.122409
$227$	1124.95	0.328924	0.164462	−	0.986383i	$-0.447411\pi$
$227$	1124.95	0.328924	0.164462	+	0.986383i	$0.447411\pi$
$228$	−1273.02	−0.369772
$229$	1931.86	0.557471	0.278735	−	0.960368i	$-0.410085\pi$
$229$	1931.86	0.557471	0.278735	+	0.960368i	$0.410085\pi$
$230$	−942.939	−0.270328
$231$	0	0
$232$	2298.46	0.650436
$233$	4282.36	1.20406	0.602031	−	0.798473i	$-0.294358\pi$
$233$	4282.36	1.20406	0.602031	+	0.798473i	$0.294358\pi$
$234$	−188.131	−0.0525577
$235$	565.366	0.156938
$236$	1173.83	0.323770
$237$	2132.64	0.584514
$238$	330.223	0.0899377
$239$	4549.61	1.23134	0.615669	−	0.788005i	$-0.288886\pi$
$239$	4549.61	1.23134	0.615669	+	0.788005i	$0.288886\pi$
$240$	−75.3836	−0.0202750
$241$	−5704.24	−1.52466	−0.762328	−	0.647191i	$-0.775944\pi$
$241$	−5704.24	−1.52466	−0.762328	+	0.647191i	$0.775944\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	−984.315	−0.258255
$245$	−576.225	−0.150260
$246$	−745.184	−0.193135
$247$	1041.62	0.268326
$248$	−923.369	−0.236427
$249$	−2706.64	−0.688860
$250$	−228.488	−0.0578033
$251$	2352.11	0.591489	0.295745	−	0.955267i	$-0.404432\pi$
$251$	2352.11	0.591489	0.295745	+	0.955267i	$0.404432\pi$
$252$	−632.774	−0.158179
$253$	0	0
$254$	−2174.72	−0.537220
$255$	−179.561	−0.0440963
$256$	−4258.05	−1.03956
$257$	−1735.01	−0.421115	−0.210558	−	0.977581i	$-0.567528\pi$
$257$	−1735.01	−0.421115	−0.210558	+	0.977581i	$0.567528\pi$
$258$	1672.13	0.403498
$259$	2251.23	0.540096
$260$	−266.383	−0.0635399
$261$	893.993	0.212018
$262$	2561.76	0.604069
$263$	3674.36	0.861485	0.430742	−	0.902475i	$-0.358252\pi$
$263$	3674.36	0.861485	0.430742	+	0.902475i	$0.358252\pi$
$264$	0	0
$265$	2764.44	0.640823
$266$	−2512.64	−0.579172
$267$	−1448.65	−0.332045
$268$	−4414.58	−1.00621
$269$	6014.78	1.36330	0.681649	−	0.731679i	$-0.261263\pi$
$269$	6014.78	1.36330	0.681649	+	0.731679i	$0.261263\pi$
$270$	−246.767	−0.0556213
$271$	3038.66	0.681128	0.340564	−	0.940221i	$-0.389382\pi$
$271$	3038.66	0.681128	0.340564	+	0.940221i	$0.389382\pi$
$272$	60.1598	0.0134107
$273$	517.750	0.114783
$274$	−1509.87	−0.332901
$275$	0	0
$276$	−1441.96	−0.314478
$277$	−5284.84	−1.14634	−0.573168	−	0.819438i	$-0.694286\pi$
$277$	−5284.84	−1.14634	−0.573168	+	0.819438i	$0.694286\pi$
$278$	3510.14	0.757282
$279$	−359.148	−0.0770668
$280$	1746.02	0.372659
$281$	6666.85	1.41534	0.707671	−	0.706542i	$-0.249746\pi$
$281$	6666.85	1.41534	0.707671	+	0.706542i	$0.249746\pi$
$282$	−620.060	−0.130936
$283$	−2563.96	−0.538557	−0.269279	−	0.963062i	$-0.586785\pi$
$283$	−2563.96	−0.538557	−0.269279	+	0.963062i	$0.586785\pi$
$284$	817.039	0.170713
$285$	1366.26	0.283967
$286$	0	0
$287$	2050.80	0.421794
$288$	−1583.33	−0.323954
$289$	−4769.70	−0.970833
$290$	−907.851	−0.183830
$291$	2172.91	0.437726
$292$	3579.28	0.717333
$293$	−5016.44	−1.00022	−0.500109	−	0.865963i	$-0.666707\pi$
$293$	−5016.44	−1.00022	−0.500109	+	0.865963i	$0.666707\pi$
$294$	631.969	0.125365
$295$	−1259.80	−0.248640
$296$	3451.68	0.677787
$297$	0	0
$298$	1932.64	0.375687
$299$	1179.85	0.228202
$300$	−349.408	−0.0672437
$301$	−4601.83	−0.881213
$302$	4101.35	0.781476
$303$	−220.580	−0.0418217
$304$	−457.751	−0.0863612
$305$	1056.41	0.198327
$306$	196.932	0.0367903
$307$	7224.25	1.34303	0.671514	−	0.740991i	$-0.265644\pi$
$307$	7224.25	1.34303	0.671514	+	0.740991i	$0.265644\pi$
$308$	0	0
$309$	−468.140	−0.0861862
$310$	364.715	0.0668207
$311$	5271.00	0.961065	0.480532	−	0.876977i	$-0.340443\pi$
$311$	5271.00	0.961065	0.480532	+	0.876977i	$0.340443\pi$
$312$	793.835	0.144045
$313$	−752.960	−0.135974	−0.0679870	−	0.997686i	$-0.521658\pi$
$313$	−752.960	−0.135974	−0.0679870	+	0.997686i	$0.521658\pi$
$314$	−699.486	−0.125714
$315$	679.120	0.121473
$316$	−3311.83	−0.589572
$317$	7792.04	1.38058	0.690291	−	0.723531i	$-0.257482\pi$
$317$	7792.04	1.38058	0.690291	+	0.723531i	$0.257482\pi$
$318$	−3031.87	−0.534651
$319$	0	0
$320$	1808.90	0.316001
$321$	−5995.85	−1.04254
$322$	−2846.08	−0.492565
$323$	−1090.35	−0.187828
$324$	−377.361	−0.0647052
$325$	285.894	0.0487955
$326$	−4595.30	−0.780705
$327$	−1926.27	−0.325758
$328$	3144.37	0.529326
$329$	1706.45	0.285957
$330$	0	0
$331$	8292.19	1.37698	0.688489	−	0.725247i	$-0.258274\pi$
$331$	8292.19	1.37698	0.688489	+	0.725247i	$0.258274\pi$
$332$	4203.21	0.694823
$333$	1342.55	0.220934
$334$	−7824.77	−1.28189
$335$	4737.91	0.772716
$336$	−227.531	−0.0369430
$337$	−925.955	−0.149674	−0.0748368	−	0.997196i	$-0.523844\pi$
$337$	−925.955	−0.149674	−0.0748368	+	0.997196i	$0.523844\pi$
$338$	3776.85	0.607792
$339$	682.564	0.109356
$340$	278.845	0.0444779
$341$	0	0
$342$	−1498.44	−0.236919
$343$	−6915.63	−1.08866
$344$	−7055.72	−1.10587
$345$	1547.58	0.241503
$346$	−4047.47	−0.628883
$347$	−2727.83	−0.422010	−0.211005	−	0.977485i	$-0.567674\pi$
$347$	−2727.83	−0.422010	−0.211005	+	0.977485i	$0.567674\pi$
$348$	−1388.31	−0.213853
$349$	−8846.92	−1.35692	−0.678459	−	0.734638i	$-0.737352\pi$
$349$	−8846.92	−1.35692	−0.678459	+	0.734638i	$0.737352\pi$
$350$	−689.647	−0.105323
$351$	308.765	0.0469535
$352$	0	0
$353$	−167.372	−0.0252360	−0.0126180	−	0.999920i	$-0.504017\pi$
$353$	−167.372	−0.0252360	−0.0126180	+	0.999920i	$0.504017\pi$
$354$	1381.68	0.207445
$355$	−876.882	−0.131099
$356$	2249.65	0.334919
$357$	−541.971	−0.0803477
$358$	−2870.97	−0.423841
$359$	10259.3	1.50826	0.754130	−	0.656725i	$-0.228059\pi$
$359$	10259.3	1.50826	0.754130	+	0.656725i	$0.228059\pi$
$360$	1041.25	0.152442
$361$	1437.35	0.209557
$362$	−5240.91	−0.760929
$363$	0	0
$364$	−804.027	−0.115776
$365$	−3841.43	−0.550876
$366$	−1158.61	−0.165468
$367$	89.0428	0.0126648	0.00633242	−	0.999980i	$-0.497984\pi$
$367$	89.0428	0.0126648	0.00633242	+	0.999980i	$0.497984\pi$
$368$	−518.497	−0.0734471
$369$	1223.02	0.172541
$370$	−1363.36	−0.191561
$371$	8343.94	1.16764
$372$	557.730	0.0777338
$373$	9532.06	1.32319	0.661597	−	0.749860i	$-0.269879\pi$
$373$	9532.06	1.32319	0.661597	+	0.749860i	$0.269879\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	2616.40	0.358858
$377$	1135.94	0.155183
$378$	−744.818	−0.101347
$379$	−8887.37	−1.20452	−0.602261	−	0.798300i	$-0.705733\pi$
$379$	−8887.37	−1.20452	−0.602261	+	0.798300i	$0.705733\pi$
$380$	−2121.71	−0.286424
$381$	3569.20	0.479936
$382$	3805.68	0.509726
$383$	2061.68	0.275057	0.137529	−	0.990498i	$-0.456084\pi$
$383$	2061.68	0.275057	0.137529	+	0.990498i	$0.456084\pi$
$384$	2238.33	0.297459
$385$	0	0
$386$	4318.75	0.569478
$387$	−2744.35	−0.360473
$388$	−3374.37	−0.441514
$389$	−7778.76	−1.01388	−0.506940	−	0.861982i	$-0.669223\pi$
$389$	−7778.76	−1.01388	−0.506940	+	0.861982i	$0.669223\pi$
$390$	−313.551	−0.0407110
$391$	−1235.04	−0.159741
$392$	−2666.65	−0.343588
$393$	−4204.43	−0.539657
$394$	−425.205	−0.0543694
$395$	3554.40	0.452762
$396$	0	0
$397$	9988.30	1.26272	0.631358	−	0.775491i	$-0.282498\pi$
$397$	9988.30	1.26272	0.631358	+	0.775491i	$0.282498\pi$
$398$	−1029.49	−0.129658
$399$	4123.81	0.517415
$400$	−125.639	−0.0157049
$401$	6331.91	0.788530	0.394265	−	0.918997i	$-0.370999\pi$
$401$	6331.91	0.788530	0.394265	+	0.918997i	$0.370999\pi$
$402$	−5196.26	−0.644691
$403$	−456.347	−0.0564077
$404$	342.544	0.0421837
$405$	405.000	0.0496904
$406$	−2740.18	−0.334957
$407$	0	0
$408$	−830.972	−0.100831
$409$	−530.445	−0.0641291	−0.0320646	−	0.999486i	$-0.510208\pi$
$409$	−530.445	−0.0641291	−0.0320646	+	0.999486i	$0.510208\pi$
$410$	−1241.97	−0.149602
$411$	2478.05	0.297404
$412$	726.987	0.0869322
$413$	−3802.48	−0.453046
$414$	−1697.29	−0.201491
$415$	−4511.06	−0.533589
$416$	−2011.84	−0.237112
$417$	−5760.94	−0.676533
$418$	0	0
$419$	575.390	0.0670875	0.0335437	−	0.999437i	$-0.489321\pi$
$419$	575.390	0.0670875	0.0335437	+	0.999437i	$0.489321\pi$
$420$	−1054.62	−0.122525
$421$	13829.7	1.60100	0.800499	−	0.599334i	$-0.204568\pi$
$421$	13829.7	1.60100	0.800499	+	0.599334i	$0.204568\pi$
$422$	−88.4466	−0.0102026
$423$	1017.66	0.116975
$424$	12793.3	1.46532
$425$	−299.268	−0.0341568
$426$	961.712	0.109378
$427$	3188.57	0.361372
$428$	9311.11	1.05156
$429$	0	0
$430$	2786.89	0.312548
$431$	4758.89	0.531850	0.265925	−	0.963994i	$-0.414323\pi$
$431$	4758.89	0.531850	0.265925	+	0.963994i	$0.414323\pi$
$432$	−135.690	−0.0151121
$433$	−4516.02	−0.501215	−0.250608	−	0.968089i	$-0.580630\pi$
$433$	−4516.02	−0.501215	−0.250608	+	0.968089i	$0.580630\pi$
$434$	1100.82	0.121754
$435$	1489.99	0.164229
$436$	2991.35	0.328578
$437$	9397.33	1.02868
$438$	4213.06	0.459606
$439$	13916.2	1.51295	0.756474	−	0.654024i	$-0.226920\pi$
$439$	13916.2	1.51295	0.756474	+	0.654024i	$0.226920\pi$
$440$	0	0
$441$	−1037.20	−0.111997
$442$	250.229	0.0269280
$443$	10294.2	1.10405	0.552023	−	0.833829i	$-0.313856\pi$
$443$	10294.2	1.10405	0.552023	+	0.833829i	$0.313856\pi$
$444$	−2084.87	−0.222846
$445$	−2414.42	−0.257201
$446$	−4319.82	−0.458631
$447$	−3171.90	−0.335628
$448$	5459.81	0.575786
$449$	−4191.42	−0.440546	−0.220273	−	0.975438i	$-0.570695\pi$
$449$	−4191.42	−0.440546	−0.220273	+	0.975438i	$0.570695\pi$
$450$	−411.278	−0.0430840
$451$	0	0
$452$	−1059.97	−0.110303
$453$	−6731.24	−0.698148
$454$	−2056.30	−0.212571
$455$	862.916	0.0889102
$456$	6322.80	0.649325
$457$	8294.55	0.849021	0.424511	−	0.905423i	$-0.360446\pi$
$457$	8294.55	0.849021	0.424511	+	0.905423i	$0.360446\pi$
$458$	−3531.24	−0.360271
$459$	−323.210	−0.0328674
$460$	−2403.27	−0.243594
$461$	1778.74	0.179706	0.0898529	−	0.995955i	$-0.471360\pi$
$461$	1778.74	0.179706	0.0898529	+	0.995955i	$0.471360\pi$
$462$	0	0
$463$	4654.76	0.467225	0.233612	−	0.972330i	$-0.424945\pi$
$463$	4654.76	0.467225	0.233612	+	0.972330i	$0.424945\pi$
$464$	−499.203	−0.0499460
$465$	−598.580	−0.0596957
$466$	−7827.72	−0.778138
$467$	−3886.35	−0.385093	−0.192547	−	0.981288i	$-0.561675\pi$
$467$	−3886.35	−0.385093	−0.192547	+	0.981288i	$0.561675\pi$
$468$	−479.490	−0.0473599
$469$	14300.5	1.40796
$470$	−1033.43	−0.101423
$471$	1148.02	0.112309
$472$	−5830.12	−0.568545
$473$	0	0
$474$	−3898.25	−0.377748
$475$	2277.11	0.219960
$476$	841.641	0.0810431
$477$	4975.99	0.477641
$478$	−8316.23	−0.795765
$479$	−11614.3	−1.10787	−0.553936	−	0.832559i	$-0.686875\pi$
$479$	−11614.3	−1.10787	−0.553936	+	0.832559i	$0.686875\pi$
$480$	−2638.89	−0.250934
$481$	1705.89	0.161709
$482$	10426.8	0.985326
$483$	4671.07	0.440043
$484$	0	0
$485$	3621.52	0.339061
$486$	−444.180	−0.0414576
$487$	−17965.5	−1.67165	−0.835826	−	0.548995i	$-0.815011\pi$
$487$	−17965.5	−1.67165	−0.835826	+	0.548995i	$0.815011\pi$
$488$	4888.85	0.453499
$489$	7541.92	0.697459
$490$	1053.28	0.0971070
$491$	15494.4	1.42414	0.712072	−	0.702107i	$-0.247757\pi$
$491$	15494.4	1.42414	0.712072	+	0.702107i	$0.247757\pi$
$492$	−1899.25	−0.174034
$493$	−1189.08	−0.108628
$494$	−1903.97	−0.173409
$495$	0	0
$496$	200.547	0.0181549
$497$	−2646.70	−0.238875
$498$	4947.47	0.445183
$499$	−15279.1	−1.37072	−0.685358	−	0.728207i	$-0.740354\pi$
$499$	−15279.1	−1.37072	−0.685358	+	0.728207i	$0.740354\pi$
$500$	−582.347	−0.0520867
$501$	12842.2	1.14521
$502$	−4299.42	−0.382256
$503$	−10613.4	−0.940810	−0.470405	−	0.882451i	$-0.655892\pi$
$503$	−10613.4	−0.940810	−0.470405	+	0.882451i	$0.655892\pi$
$504$	3142.83	0.277763
$505$	−367.633	−0.0323950
$506$	0	0
$507$	−6198.67	−0.542983
$508$	−5542.71	−0.484090
$509$	15287.8	1.33127	0.665637	−	0.746276i	$-0.268160\pi$
$509$	15287.8	1.33127	0.665637	+	0.746276i	$0.268160\pi$
$510$	328.220	0.0284977
$511$	−11594.6	−1.00375
$512$	1814.42	0.156615
$513$	2459.28	0.211656
$514$	3171.42	0.272150
$515$	−780.233	−0.0667596
$516$	4261.77	0.363593
$517$	0	0
$518$	−4115.03	−0.349043
$519$	6642.82	0.561825
$520$	1323.06	0.111577
$521$	9592.74	0.806652	0.403326	−	0.915056i	$-0.367854\pi$
$521$	9592.74	0.806652	0.403326	+	0.915056i	$0.367854\pi$
$522$	−1634.13	−0.137019
$523$	−4245.93	−0.354994	−0.177497	−	0.984121i	$-0.556800\pi$
$523$	−4245.93	−0.354994	−0.177497	+	0.984121i	$0.556800\pi$
$524$	6529.16	0.544328
$525$	1131.87	0.0940928
$526$	−6716.36	−0.556744
$527$	477.696	0.0394853
$528$	0	0
$529$	−1522.59	−0.125141
$530$	−5053.12	−0.414139
$531$	−2267.65	−0.185325
$532$	−6403.97	−0.521894
$533$	1554.01	0.126288
$534$	2647.99	0.214588
$535$	−9993.08	−0.807549
$536$	21926.1	1.76691
$537$	4711.91	0.378648
$538$	−10994.4	−0.881046
$539$	0	0
$540$	−628.935	−0.0501205
$541$	14877.6	1.18232	0.591161	−	0.806554i	$-0.298670\pi$
$541$	14877.6	1.18232	0.591161	+	0.806554i	$0.298670\pi$
$542$	−5554.38	−0.440186
$543$	8601.53	0.679792
$544$	2105.96	0.165978
$545$	−3210.45	−0.252331
$546$	−946.395	−0.0741795
$547$	22641.9	1.76983	0.884915	−	0.465753i	$-0.154216\pi$
$547$	22641.9	1.76983	0.884915	+	0.465753i	$0.154216\pi$
$548$	−3848.22	−0.299978
$549$	1901.54	0.147824
$550$	0	0
$551$	9047.64	0.699533
$552$	7161.87	0.552227
$553$	10728.3	0.824977
$554$	9660.16	0.740832
$555$	2237.58	0.171135
$556$	8946.31	0.682389
$557$	9738.83	0.740840	0.370420	−	0.928864i	$-0.379214\pi$
$557$	9738.83	0.740840	0.370420	+	0.928864i	$0.379214\pi$
$558$	656.487	0.0498052
$559$	−3487.08	−0.263842
$560$	−379.219	−0.0286159
$561$	0	0
$562$	−12186.3	−0.914680
$563$	8888.54	0.665377	0.332688	−	0.943037i	$-0.392044\pi$
$563$	8888.54	0.665377	0.332688	+	0.943037i	$0.392044\pi$
$564$	−1580.35	−0.117987
$565$	1137.61	0.0847071
$566$	4686.67	0.348048
$567$	1222.42	0.0905408
$568$	−4058.03	−0.299773
$569$	−12443.0	−0.916764	−0.458382	−	0.888755i	$-0.651571\pi$
$569$	−12443.0	−0.916764	−0.458382	+	0.888755i	$0.651571\pi$
$570$	−2497.40	−0.183517
$571$	−17508.4	−1.28320	−0.641598	−	0.767041i	$-0.721728\pi$
$571$	−17508.4	−1.28320	−0.641598	+	0.767041i	$0.721728\pi$
$572$	0	0
$573$	−6245.98	−0.455375
$574$	−3748.66	−0.272589
$575$	2579.29	0.187068
$576$	3256.01	0.235533
$577$	−26750.4	−1.93004	−0.965021	−	0.262172i	$-0.915561\pi$
$577$	−26750.4	−1.93004	−0.965021	+	0.262172i	$0.915561\pi$
$578$	8718.54	0.627411
$579$	−7088.05	−0.508755
$580$	−2313.84	−0.165650
$581$	−13615.8	−0.972252
$582$	−3971.86	−0.282885
$583$	0	0
$584$	−17777.4	−1.25965
$585$	514.609	0.0363700
$586$	9169.56	0.646401
$587$	16631.8	1.16945	0.584727	−	0.811230i	$-0.301202\pi$
$587$	16631.8	1.16945	0.584727	+	0.811230i	$0.301202\pi$
$588$	1610.70	0.112966
$589$	−3634.75	−0.254274
$590$	2302.80	0.160686
$591$	697.858	0.0485720
$592$	−749.673	−0.0520463
$593$	4248.11	0.294181	0.147090	−	0.989123i	$-0.453009\pi$
$593$	4248.11	0.294181	0.147090	+	0.989123i	$0.453009\pi$
$594$	0	0
$595$	−903.285	−0.0622371
$596$	4925.73	0.338533
$597$	1689.63	0.115833
$598$	−2156.64	−0.147478
$599$	−22152.5	−1.51106	−0.755531	−	0.655113i	$-0.772621\pi$
$599$	−22152.5	−1.51106	−0.755531	+	0.655113i	$0.772621\pi$
$600$	1735.42	0.118081
$601$	−1870.46	−0.126951	−0.0634756	−	0.997983i	$-0.520218\pi$
$601$	−1870.46	−0.126951	−0.0634756	+	0.997983i	$0.520218\pi$
$602$	8411.69	0.569493
$603$	8528.24	0.575948
$604$	10453.1	0.704190
$605$	0	0
$606$	403.198	0.0270277
$607$	−17092.6	−1.14294	−0.571472	−	0.820622i	$-0.693627\pi$
$607$	−17092.6	−1.14294	−0.571472	+	0.820622i	$0.693627\pi$
$608$	−16024.1	−1.06885
$609$	4497.25	0.299241
$610$	−1931.01	−0.128171
$611$	1293.08	0.0856176
$612$	501.921	0.0331519
$613$	−29572.2	−1.94846	−0.974232	−	0.225550i	$-0.927582\pi$
$613$	−29572.2	−1.94846	−0.974232	+	0.225550i	$0.927582\pi$
$614$	−13205.2	−0.867947
$615$	2038.36	0.133650
$616$	0	0
$617$	−10496.0	−0.684854	−0.342427	−	0.939544i	$-0.611249\pi$
$617$	−10496.0	−0.684854	−0.342427	+	0.939544i	$0.611249\pi$
$618$	855.713	0.0556988
$619$	−2943.78	−0.191148	−0.0955740	−	0.995422i	$-0.530469\pi$
$619$	−2943.78	−0.191148	−0.0955740	+	0.995422i	$0.530469\pi$
$620$	929.550	0.0602123
$621$	2785.64	0.180006
$622$	−9634.87	−0.621098
$623$	−7287.47	−0.468646
$624$	−172.414	−0.0110610
$625$	625.000	0.0400000
$626$	1376.34	0.0878746
$627$	0	0
$628$	−1782.78	−0.113281
$629$	−1785.69	−0.113196
$630$	−1241.36	−0.0785034
$631$	15591.4	0.983653	0.491827	−	0.870693i	$-0.336329\pi$
$631$	15591.4	0.983653	0.491827	+	0.870693i	$0.336329\pi$
$632$	16449.0	1.03530
$633$	145.161	0.00911474
$634$	−14243.1	−0.892216
$635$	5948.67	0.371757
$636$	−7727.34	−0.481775
$637$	−1317.91	−0.0819743
$638$	0	0
$639$	−1578.39	−0.0977152
$640$	3730.54	0.230410
$641$	22162.7	1.36564	0.682820	−	0.730586i	$-0.260753\pi$
$641$	22162.7	1.36564	0.682820	+	0.730586i	$0.260753\pi$
$642$	10959.8	0.673753
$643$	−24199.0	−1.48416	−0.742081	−	0.670310i	$-0.766161\pi$
$643$	−24199.0	−1.48416	−0.742081	+	0.670310i	$0.766161\pi$
$644$	−7253.82	−0.443852
$645$	−4573.91	−0.279221
$646$	1993.04	0.121386
$647$	23519.5	1.42913	0.714564	−	0.699570i	$-0.246625\pi$
$647$	23519.5	1.42913	0.714564	+	0.699570i	$0.246625\pi$
$648$	1874.26	0.113623
$649$	0	0
$650$	−522.586	−0.0315346
$651$	−1806.70	−0.108771
$652$	−11712.0	−0.703496
$653$	10747.9	0.644099	0.322050	−	0.946723i	$-0.395628\pi$
$653$	10747.9	0.644099	0.322050	+	0.946723i	$0.395628\pi$
$654$	3521.03	0.210525
$655$	−7007.38	−0.418017
$656$	−682.928	−0.0406461
$657$	−6914.58	−0.410599
$658$	−3119.22	−0.184802
$659$	−29529.6	−1.74554	−0.872769	−	0.488133i	$-0.837678\pi$
$659$	−29529.6	−1.74554	−0.872769	+	0.488133i	$0.837678\pi$
$660$	0	0
$661$	33178.7	1.95235	0.976175	−	0.216984i	$-0.0696218\pi$
$661$	33178.7	1.95235	0.976175	+	0.216984i	$0.0696218\pi$
$662$	−15157.3	−0.889887
$663$	−410.683	−0.0240567
$664$	−20876.3	−1.22012
$665$	6873.02	0.400788
$666$	−2454.04	−0.142781
$667$	10248.3	0.594927
$668$	−19943.0	−1.15512
$669$	7089.80	0.409727
$670$	−8660.43	−0.499376
$671$	0	0
$672$	−7964.98	−0.457226
$673$	−18591.2	−1.06484	−0.532419	−	0.846481i	$-0.678717\pi$
$673$	−18591.2	−1.06484	−0.532419	+	0.846481i	$0.678717\pi$
$674$	1692.55	0.0967281
$675$	675.000	0.0384900
$676$	9626.08	0.547683
$677$	5323.51	0.302214	0.151107	−	0.988517i	$-0.451716\pi$
$677$	5323.51	0.302214	0.151107	+	0.988517i	$0.451716\pi$
$678$	−1247.66	−0.0706727
$679$	10930.9	0.617802
$680$	−1384.95	−0.0781037
$681$	3374.86	0.189904
$682$	0	0
$683$	−19258.0	−1.07890	−0.539450	−	0.842018i	$-0.681368\pi$
$683$	−19258.0	−1.07890	−0.539450	+	0.842018i	$0.681368\pi$
$684$	−3819.07	−0.213488
$685$	4130.08	0.230368
$686$	12641.1	0.703555
$687$	5795.57	0.321856
$688$	1532.44	0.0849180
$689$	6322.69	0.349601
$690$	−2828.82	−0.156074
$691$	−25193.1	−1.38696	−0.693481	−	0.720475i	$-0.743924\pi$
$691$	−25193.1	−1.38696	−0.693481	+	0.720475i	$0.743924\pi$
$692$	−10315.8	−0.566688
$693$	0	0
$694$	4986.20	0.272729
$695$	−9601.56	−0.524040
$696$	6895.37	0.375529
$697$	−1626.71	−0.0884018
$698$	16171.3	0.876923
$699$	12847.1	0.695166
$700$	−1757.70	−0.0949071
$701$	−16986.7	−0.915233	−0.457617	−	0.889150i	$-0.651297\pi$
$701$	−16986.7	−0.915233	−0.457617	+	0.889150i	$0.651297\pi$
$702$	−564.392	−0.0303442
$703$	13587.2	0.728949
$704$	0	0
$705$	1696.10	0.0906082
$706$	305.940	0.0163090
$707$	−1109.63	−0.0590268
$708$	3521.49	0.186929
$709$	−27674.9	−1.46594	−0.732972	−	0.680259i	$-0.761867\pi$
$709$	−27674.9	−1.46594	−0.732972	+	0.680259i	$0.761867\pi$
$710$	1602.85	0.0847239
$711$	6397.91	0.337469
$712$	−11173.4	−0.588122
$713$	−4117.10	−0.216251
$714$	990.669	0.0519256
$715$	0	0
$716$	−7317.24	−0.381925
$717$	13648.8	0.710913
$718$	−18753.0	−0.974729
$719$	34365.9	1.78252	0.891260	−	0.453494i	$-0.149822\pi$
$719$	34365.9	1.78252	0.891260	+	0.453494i	$0.149822\pi$
$720$	−226.151	−0.0117058
$721$	−2354.99	−0.121643
$722$	−2627.33	−0.135428
$723$	−17112.7	−0.880261
$724$	−13357.5	−0.685675
$725$	2483.31	0.127211
$726$	0	0
$727$	−10929.0	−0.557543	−0.278772	−	0.960357i	$-0.589927\pi$
$727$	−10929.0	−0.557543	−0.278772	+	0.960357i	$0.589927\pi$
$728$	3993.40	0.203304
$729$	729.000	0.0370370
$730$	7021.76	0.356010
$731$	3650.21	0.184689
$732$	−2952.94	−0.149104
$733$	−34482.0	−1.73755	−0.868774	−	0.495209i	$-0.835092\pi$
$733$	−34482.0	−1.73755	−0.868774	+	0.495209i	$0.835092\pi$
$734$	−162.761	−0.00818478
$735$	−1728.67	−0.0867526
$736$	−18150.6	−0.909020
$737$	0	0
$738$	−2235.55	−0.111506
$739$	20970.7	1.04387	0.521935	−	0.852985i	$-0.325210\pi$
$739$	20970.7	1.04387	0.521935	+	0.852985i	$0.325210\pi$
$740$	−3474.79	−0.172616
$741$	3124.85	0.154918
$742$	−15251.9	−0.754602
$743$	−23789.1	−1.17461	−0.587306	−	0.809365i	$-0.699812\pi$
$743$	−23789.1	−1.17461	−0.587306	+	0.809365i	$0.699812\pi$
$744$	−2770.11	−0.136501
$745$	−5286.50	−0.259976
$746$	−17423.7	−0.855128
$747$	−8119.92	−0.397714
$748$	0	0
$749$	−30162.2	−1.47143
$750$	−685.463	−0.0333728
$751$	−16312.5	−0.792610	−0.396305	−	0.918119i	$-0.629708\pi$
$751$	−16312.5	−0.792610	−0.396305	+	0.918119i	$0.629708\pi$
$752$	−568.258	−0.0275562
$753$	7056.33	0.341496
$754$	−2076.39	−0.100289
$755$	−11218.7	−0.540783
$756$	−1898.32	−0.0913244
$757$	−13835.0	−0.664258	−0.332129	−	0.943234i	$-0.607767\pi$
$757$	−13835.0	−0.664258	−0.332129	+	0.943234i	$0.607767\pi$
$758$	16245.2	0.778435
$759$	0	0
$760$	10538.0	0.502965
$761$	27153.6	1.29345	0.646727	−	0.762722i	$-0.276137\pi$
$761$	27153.6	1.29345	0.646727	+	0.762722i	$0.276137\pi$
$762$	−6524.15	−0.310164
$763$	−9690.13	−0.459772
$764$	9699.55	0.459316
$765$	−538.683	−0.0254590
$766$	−3768.55	−0.177759
$767$	−2881.36	−0.135645
$768$	−12774.1	−0.600192
$769$	−4108.08	−0.192641	−0.0963207	−	0.995350i	$-0.530707\pi$
$769$	−4108.08	−0.192641	−0.0963207	+	0.995350i	$0.530707\pi$
$770$	0	0
$771$	−5205.02	−0.243131
$772$	11007.2	0.513158
$773$	−8280.36	−0.385283	−0.192642	−	0.981269i	$-0.561705\pi$
$773$	−8280.36	−0.385283	−0.192642	+	0.981269i	$0.561705\pi$
$774$	5016.40	0.232960
$775$	−997.633	−0.0462401
$776$	16759.6	0.775304
$777$	6753.70	0.311825
$778$	14218.8	0.655230
$779$	12377.5	0.569281
$780$	−799.149	−0.0366848
$781$	0	0
$782$	2257.53	0.103234
$783$	2681.98	0.122409
$784$	579.172	0.0263836
$785$	1913.36	0.0869945
$786$	7685.28	0.348759
$787$	5701.06	0.258222	0.129111	−	0.991630i	$-0.458788\pi$
$787$	5701.06	0.258222	0.129111	+	0.991630i	$0.458788\pi$
$788$	−1083.72	−0.0489924
$789$	11023.1	0.497378
$790$	−6497.08	−0.292602
$791$	3433.65	0.154345
$792$	0	0
$793$	2416.17	0.108197
$794$	−18257.6	−0.816044
$795$	8293.32	0.369979
$796$	−2623.87	−0.116835
$797$	−12921.0	−0.574262	−0.287131	−	0.957891i	$-0.592702\pi$
$797$	−12921.0	−0.574262	−0.287131	+	0.957891i	$0.592702\pi$
$798$	−7537.92	−0.334385
$799$	−1353.57	−0.0599322
$800$	−4398.14	−0.194372
$801$	−4345.96	−0.191706
$802$	−11574.1	−0.509596
$803$	0	0
$804$	−13243.7	−0.580933
$805$	7785.11	0.340856
$806$	834.158	0.0364540
$807$	18044.3	0.787101
$808$	−1701.33	−0.0740751
$809$	−19951.1	−0.867048	−0.433524	−	0.901142i	$-0.642730\pi$
$809$	−19951.1	−0.867048	−0.433524	+	0.901142i	$0.642730\pi$
$810$	−740.300	−0.0321129
$811$	4604.36	0.199360	0.0996800	−	0.995020i	$-0.468218\pi$
$811$	4604.36	0.199360	0.0996800	+	0.995020i	$0.468218\pi$
$812$	−6983.90	−0.301831
$813$	9115.99	0.393249
$814$	0	0
$815$	12569.9	0.540250
$816$	180.479	0.00774270
$817$	−27774.1	−1.18934
$818$	969.601	0.0414441
$819$	1553.25	0.0662698
$820$	−3165.42	−0.134806
$821$	6932.84	0.294711	0.147356	−	0.989084i	$-0.452924\pi$
$821$	6932.84	0.294711	0.147356	+	0.989084i	$0.452924\pi$
$822$	−4529.62	−0.192200
$823$	−23247.6	−0.984642	−0.492321	−	0.870414i	$-0.663851\pi$
$823$	−23247.6	−0.984642	−0.492321	+	0.870414i	$0.663851\pi$
$824$	−3610.76	−0.152654
$825$	0	0
$826$	6950.56	0.292786
$827$	30227.9	1.27101	0.635505	−	0.772097i	$-0.280792\pi$
$827$	30227.9	1.27101	0.635505	+	0.772097i	$0.280792\pi$
$828$	−4325.89	−0.181564
$829$	22612.6	0.947366	0.473683	−	0.880695i	$-0.342924\pi$
$829$	22612.6	0.947366	0.473683	+	0.880695i	$0.342924\pi$
$830$	8245.78	0.344838
$831$	−15854.5	−0.661838
$832$	4137.22	0.172395
$833$	1379.57	0.0573819
$834$	10530.4	0.437217
$835$	21403.7	0.887073
$836$	0	0
$837$	−1077.44	−0.0444945
$838$	−1051.76	−0.0433560
$839$	−36758.8	−1.51258	−0.756290	−	0.654236i	$-0.772990\pi$
$839$	−36758.8	−1.51258	−0.756290	+	0.654236i	$0.772990\pi$
$840$	5238.05	0.215155
$841$	−14522.0	−0.595434
$842$	−25279.4	−1.03466
$843$	20000.6	0.817148
$844$	−225.424	−0.00919363
$845$	−10331.1	−0.420593
$846$	−1860.18	−0.0755961
$847$	0	0
$848$	−2778.58	−0.112520
$849$	−7691.88	−0.310936
$850$	547.033	0.0220742
$851$	15390.3	0.619945
$852$	2451.12	0.0985609
$853$	−16399.4	−0.658269	−0.329135	−	0.944283i	$-0.606757\pi$
$853$	−16399.4	−0.658269	−0.329135	+	0.944283i	$0.606757\pi$
$854$	−5828.39	−0.233540
$855$	4098.79	0.163948
$856$	−46246.0	−1.84656
$857$	9966.88	0.397272	0.198636	−	0.980073i	$-0.436349\pi$
$857$	9966.88	0.397272	0.198636	+	0.980073i	$0.436349\pi$
$858$	0	0
$859$	1074.79	0.0426907	0.0213453	−	0.999772i	$-0.493205\pi$
$859$	1074.79	0.0426907	0.0213453	+	0.999772i	$0.493205\pi$
$860$	7102.95	0.281638
$861$	6152.40	0.243523
$862$	−8698.77	−0.343714
$863$	−23860.5	−0.941159	−0.470580	−	0.882357i	$-0.655955\pi$
$863$	−23860.5	−0.941159	−0.470580	+	0.882357i	$0.655955\pi$
$864$	−4749.99	−0.187035
$865$	11071.4	0.435188
$866$	8254.84	0.323916
$867$	−14309.1	−0.560511
$868$	2805.67	0.109713
$869$	0	0
$870$	−2723.55	−0.106135
$871$	10836.3	0.421555
$872$	−14857.3	−0.576986
$873$	6518.73	0.252721
$874$	−17177.4	−0.664798
$875$	1886.44	0.0728839
$876$	10737.8	0.414153
$877$	−11183.2	−0.430591	−0.215295	−	0.976549i	$-0.569071\pi$
$877$	−11183.2	−0.430591	−0.215295	+	0.976549i	$0.569071\pi$
$878$	−25437.4	−0.977759
$879$	−15049.3	−0.577476
$880$	0	0
$881$	20779.5	0.794642	0.397321	−	0.917680i	$-0.369940\pi$
$881$	20779.5	0.794642	0.397321	+	0.917680i	$0.369940\pi$
$882$	1895.91	0.0723793
$883$	−6724.99	−0.256301	−0.128150	−	0.991755i	$-0.540904\pi$
$883$	−6724.99	−0.256301	−0.128150	+	0.991755i	$0.540904\pi$
$884$	637.760	0.0242649
$885$	−3779.41	−0.143552
$886$	−18816.8	−0.713501
$887$	14957.5	0.566205	0.283102	−	0.959090i	$-0.408636\pi$
$887$	14957.5	0.566205	0.283102	+	0.959090i	$0.408636\pi$
$888$	10355.1	0.391321
$889$	17954.9	0.677378
$890$	4413.32	0.166219
$891$	0	0
$892$	−11009.9	−0.413274
$893$	10299.2	0.385946
$894$	5797.92	0.216903
$895$	7853.18	0.293299
$896$	11259.9	0.419831
$897$	3539.54	0.131752
$898$	7661.50	0.284708
$899$	−3963.90	−0.147056
$900$	−1048.22	−0.0388231
$901$	−6618.47	−0.244721
$902$	0	0
$903$	−13805.5	−0.508769
$904$	5264.62	0.193693
$905$	14335.9	0.526564
$906$	12304.0	0.451186
$907$	−52411.3	−1.91873	−0.959366	−	0.282164i	$-0.908948\pi$
$907$	−52411.3	−1.91873	−0.959366	+	0.282164i	$0.908948\pi$
$908$	−5240.91	−0.191548
$909$	−661.740	−0.0241458
$910$	−1577.33	−0.0574592
$911$	−19174.4	−0.697339	−0.348670	−	0.937246i	$-0.613366\pi$
$911$	−19174.4	−0.697339	−0.348670	+	0.937246i	$0.613366\pi$
$912$	−1373.25	−0.0498607
$913$	0	0
$914$	−15161.6	−0.548689
$915$	3169.23	0.114504
$916$	−9000.09	−0.324641
$917$	−21150.4	−0.761667
$918$	590.795	0.0212409
$919$	18396.3	0.660325	0.330162	−	0.943924i	$-0.392896\pi$
$919$	18396.3	0.660325	0.330162	+	0.943924i	$0.392896\pi$
$920$	11936.5	0.427753
$921$	21672.8	0.775398
$922$	−3251.37	−0.116137
$923$	−2005.56	−0.0715209
$924$	0	0
$925$	3729.29	0.132560
$926$	−8508.44	−0.301949
$927$	−1404.42	−0.0497597
$928$	−17475.2	−0.618157
$929$	−17244.1	−0.608998	−0.304499	−	0.952513i	$-0.598489\pi$
$929$	−17244.1	−0.608998	−0.304499	+	0.952513i	$0.598489\pi$
$930$	1094.14	0.0385790
$931$	−10497.0	−0.369523
$932$	−19950.5	−0.701182
$933$	15813.0	0.554871
$934$	7103.86	0.248871
$935$	0	0
$936$	2381.51	0.0831645
$937$	8269.68	0.288323	0.144162	−	0.989554i	$-0.453952\pi$
$937$	8269.68	0.288323	0.144162	+	0.989554i	$0.453952\pi$
$938$	−26139.9	−0.909912
$939$	−2258.88	−0.0785046
$940$	−2633.92	−0.0913924
$941$	55133.8	1.91000	0.955000	−	0.296606i	$-0.0958548\pi$
$941$	55133.8	1.91000	0.955000	+	0.296606i	$0.0958548\pi$
$942$	−2098.46	−0.0725812
$943$	14020.1	0.484153
$944$	1266.25	0.0436577
$945$	2037.36	0.0701326
$946$	0	0
$947$	−49363.9	−1.69389	−0.846944	−	0.531682i	$-0.821560\pi$
$947$	−49363.9	−1.69389	−0.846944	+	0.531682i	$0.821560\pi$
$948$	−9935.48	−0.340390
$949$	−8785.94	−0.300531
$950$	−4162.33	−0.142151
$951$	23376.1	0.797080
$952$	−4180.22	−0.142313
$953$	−44934.5	−1.52736	−0.763678	−	0.645597i	$-0.776609\pi$
$953$	−44934.5	−1.52736	−0.763678	+	0.645597i	$0.776609\pi$
$954$	−9095.62	−0.308681
$955$	−10410.0	−0.352732
$956$	−21195.6	−0.717066
$957$	0	0
$958$	21229.8	0.715975
$959$	12465.9	0.419753
$960$	5426.69	0.182443
$961$	−28198.6	−0.946546
$962$	−3118.20	−0.104506
$963$	−17987.5	−0.601911
$964$	26574.8	0.887879
$965$	−11813.4	−0.394080
$966$	−8538.25	−0.284383
$967$	−258.493	−0.00859625	−0.00429813	−	0.999991i	$-0.501368\pi$
$967$	−258.493	−0.00859625	−0.00429813	+	0.999991i	$0.501368\pi$
$968$	0	0
$969$	−3271.04	−0.108443
$970$	−6619.77	−0.219122
$971$	47477.2	1.56912	0.784561	−	0.620052i	$-0.212889\pi$
$971$	47477.2	1.56912	0.784561	+	0.620052i	$0.212889\pi$
$972$	−1132.08	−0.0373576
$973$	−28980.5	−0.954853
$974$	32839.2	1.08032
$975$	857.681	0.0281721
$976$	−1061.81	−0.0348235
$977$	27864.9	0.912463	0.456231	−	0.889861i	$-0.349199\pi$
$977$	27864.9	0.912463	0.456231	+	0.889861i	$0.349199\pi$
$978$	−13785.9	−0.450740
$979$	0	0
$980$	2684.50	0.0875034
$981$	−5778.81	−0.188077
$982$	−28322.3	−0.920368
$983$	8177.20	0.265323	0.132661	−	0.991161i	$-0.457648\pi$
$983$	8177.20	0.265323	0.132661	+	0.991161i	$0.457648\pi$
$984$	9433.12	0.305606
$985$	1163.10	0.0376237
$986$	2173.53	0.0702021
$987$	5119.35	0.165097
$988$	−4852.66	−0.156259
$989$	−31459.9	−1.01149
$990$	0	0
$991$	51787.1	1.66001	0.830005	−	0.557756i	$-0.188337\pi$
$991$	51787.1	1.66001	0.830005	+	0.557756i	$0.188337\pi$
$992$	7020.37	0.224695
$993$	24876.6	0.794999
$994$	4837.91	0.154375
$995$	2816.05	0.0897235
$996$	12609.6	0.401156
$997$	22176.1	0.704439	0.352219	−	0.935917i	$-0.385427\pi$
$997$	22176.1	0.704439	0.352219	+	0.935917i	$0.385427\pi$
$998$	27928.7	0.885839
$999$	4027.64	0.127556

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bb.1.3	✓	6
11.10	odd	2	inner	1815.4.a.bb.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.4.a.bb.1.3	✓	6	1.1	even	1	trivial
1815.4.a.bb.1.4	yes	6	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82790 q^{2} +3.00000 q^{3} -4.65878 q^{4} +5.00000 q^{5} -5.48370 q^{6} +15.0916 q^{7} +23.1390 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.82790 q^{2} +3.00000 q^{3} -4.65878 q^{4} +5.00000 q^{5} -5.48370 q^{6} +15.0916 q^{7} +23.1390 q^{8} +9.00000 q^{9} -9.13951 q^{10} -13.9763 q^{12} +11.4358 q^{13} -27.5859 q^{14} +15.0000 q^{15} -5.02557 q^{16} -11.9707 q^{17} -16.4511 q^{18} +91.0843 q^{19} -23.2939 q^{20} +45.2747 q^{21} +103.172 q^{23} +69.4170 q^{24} +25.0000 q^{25} -20.9034 q^{26} +27.0000 q^{27} -70.3082 q^{28} +99.3326 q^{29} -27.4185 q^{30} -39.9053 q^{31} -175.926 q^{32} +21.8813 q^{34} +75.4578 q^{35} -41.9290 q^{36} +149.172 q^{37} -166.493 q^{38} +34.3073 q^{39} +115.695 q^{40} +135.891 q^{41} -82.7576 q^{42} -304.928 q^{43} +45.0000 q^{45} -188.588 q^{46} +113.073 q^{47} -15.0767 q^{48} -115.245 q^{49} -45.6975 q^{50} -35.9122 q^{51} -53.2766 q^{52} +552.888 q^{53} -49.3533 q^{54} +349.203 q^{56} +273.253 q^{57} -181.570 q^{58} -251.961 q^{59} -69.8817 q^{60} +211.282 q^{61} +72.9430 q^{62} +135.824 q^{63} +361.779 q^{64} +57.1788 q^{65} +947.582 q^{67} +55.7690 q^{68} +309.515 q^{69} -137.929 q^{70} -175.376 q^{71} +208.251 q^{72} -768.287 q^{73} -272.671 q^{74} +75.0000 q^{75} -424.342 q^{76} -62.7103 q^{78} +710.879 q^{79} -25.1279 q^{80} +81.0000 q^{81} -248.395 q^{82} -902.213 q^{83} -210.925 q^{84} -59.8537 q^{85} +557.377 q^{86} +297.998 q^{87} -482.884 q^{89} -82.2556 q^{90} +172.583 q^{91} -480.654 q^{92} -119.716 q^{93} -206.687 q^{94} +455.422 q^{95} -527.777 q^{96} +724.303 q^{97} +210.656 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10}) \) 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9	\( 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9} + 120 q^{12} + 116 q^{14} + 90 q^{15} + 164 q^{16} + 200 q^{20} + 56 q^{23} + 150 q^{25} + 292 q^{26} + 162 q^{27} + 576 q^{31} - 92 q^{34} + 360 q^{36} + 332 q^{37} + 496 q^{38} + 348 q^{42} + 270 q^{45} + 96 q^{47} + 492 q^{48} + 454 q^{49} + 308 q^{53} - 652 q^{56} + 1784 q^{58} + 2080 q^{59} + 600 q^{60} + 1928 q^{64} - 1168 q^{67} + 168 q^{69} + 580 q^{70} + 1064 q^{71} + 450 q^{75} + 876 q^{78} + 820 q^{80} + 486 q^{81} + 24 q^{82} + 5412 q^{86} - 684 q^{89} + 2744 q^{91} + 1368 q^{92} + 1728 q^{93} + 2812 q^{97}+O(q^{100}) \) 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9 + 120 * q^12 + 116 * q^14 + 90 * q^15 + 164 * q^16 + 200 * q^20 + 56 * q^23 + 150 * q^25 + 292 * q^26 + 162 * q^27 + 576 * q^31 - 92 * q^34 + 360 * q^36 + 332 * q^37 + 496 * q^38 + 348 * q^42 + 270 * q^45 + 96 * q^47 + 492 * q^48 + 454 * q^49 + 308 * q^53 - 652 * q^56 + 1784 * q^58 + 2080 * q^59 + 600 * q^60 + 1928 * q^64 - 1168 * q^67 + 168 * q^69 + 580 * q^70 + 1064 * q^71 + 450 * q^75 + 876 * q^78 + 820 * q^80 + 486 * q^81 + 24 * q^82 + 5412 * q^86 - 684 * q^89 + 2744 * q^91 + 1368 * q^92 + 1728 * q^93 + 2812 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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