LMFDB - Embedded newform 1815.4.a.v.1.2

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.744012.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 21x^{2} + 48 $ x^4 - 21*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.61559$ 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)

$f(q)$	$=$	$q-1.61559 q^{2} -3.00000 q^{3} -5.38987 q^{4} +5.00000 q^{5} +4.84677 q^{6} -11.6241 q^{7} +21.6325 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.61559 q^{2} -3.00000 q^{3} -5.38987 q^{4} +5.00000 q^{5} +4.84677 q^{6} -11.6241 q^{7} +21.6325 q^{8} +9.00000 q^{9} -8.07795 q^{10} +16.1696 q^{12} -54.8892 q^{13} +18.7797 q^{14} -15.0000 q^{15} +8.16960 q^{16} +81.3685 q^{17} -14.5403 q^{18} +22.6183 q^{19} -26.9493 q^{20} +34.8722 q^{21} -150.238 q^{23} -64.8976 q^{24} +25.0000 q^{25} +88.6784 q^{26} -27.0000 q^{27} +62.6522 q^{28} -18.7572 q^{29} +24.2339 q^{30} -8.00000 q^{31} -186.259 q^{32} -131.458 q^{34} -58.1203 q^{35} -48.5088 q^{36} +113.797 q^{37} -36.5419 q^{38} +164.667 q^{39} +108.163 q^{40} -208.521 q^{41} -56.3392 q^{42} +182.083 q^{43} +45.0000 q^{45} +242.723 q^{46} +413.154 q^{47} -24.5088 q^{48} -207.881 q^{49} -40.3898 q^{50} -244.105 q^{51} +295.845 q^{52} +222.476 q^{53} +43.6209 q^{54} -251.458 q^{56} -67.8548 q^{57} +30.3040 q^{58} +55.7973 q^{59} +80.8480 q^{60} +233.823 q^{61} +12.9247 q^{62} -104.617 q^{63} +235.562 q^{64} -274.446 q^{65} +274.238 q^{67} -438.565 q^{68} +450.714 q^{69} +93.8987 q^{70} -39.7269 q^{71} +194.693 q^{72} +623.250 q^{73} -183.850 q^{74} -75.0000 q^{75} -121.909 q^{76} -266.035 q^{78} +152.798 q^{79} +40.8480 q^{80} +81.0000 q^{81} +336.885 q^{82} -992.373 q^{83} -187.957 q^{84} +406.842 q^{85} -294.172 q^{86} +56.2717 q^{87} -407.154 q^{89} -72.7016 q^{90} +638.035 q^{91} +809.762 q^{92} +24.0000 q^{93} -667.488 q^{94} +113.091 q^{95} +558.777 q^{96} -602.678 q^{97} +335.851 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 10 q^{4} + 20 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 10 * q^4 + 20 * q^5 + 36 * q^9	$ 4 q - 12 q^{3} + 10 q^{4} + 20 q^{5} + 36 q^{9} - 30 q^{12} + 12 q^{14} - 60 q^{15} - 62 q^{16} + 50 q^{20} - 96 q^{23} + 100 q^{25} - 24 q^{26} - 108 q^{27} - 32 q^{31} - 84 q^{34} + 90 q^{36} - 176 q^{37} - 588 q^{38} - 36 q^{42} + 180 q^{45} + 264 q^{47} + 186 q^{48} - 1084 q^{49} - 120 q^{53} - 564 q^{56} + 1068 q^{58} - 408 q^{59} - 150 q^{60} - 1046 q^{64} + 592 q^{67} + 288 q^{69} + 60 q^{70} - 1800 q^{71} - 300 q^{75} + 72 q^{78} - 310 q^{80} + 324 q^{81} - 2124 q^{82} + 780 q^{86} - 240 q^{89} + 1416 q^{91} + 3744 q^{92} + 96 q^{93} - 2032 q^{97}+O(q^{100}) $ 4 * q - 12 * q^3 + 10 * q^4 + 20 * q^5 + 36 * q^9 - 30 * q^12 + 12 * q^14 - 60 * q^15 - 62 * q^16 + 50 * q^20 - 96 * q^23 + 100 * q^25 - 24 * q^26 - 108 * q^27 - 32 * q^31 - 84 * q^34 + 90 * q^36 - 176 * q^37 - 588 * q^38 - 36 * q^42 + 180 * q^45 + 264 * q^47 + 186 * q^48 - 1084 * q^49 - 120 * q^53 - 564 * q^56 + 1068 * q^58 - 408 * q^59 - 150 * q^60 - 1046 * q^64 + 592 * q^67 + 288 * q^69 + 60 * q^70 - 1800 * q^71 - 300 * q^75 + 72 * q^78 - 310 * q^80 + 324 * q^81 - 2124 * q^82 + 780 * q^86 - 240 * q^89 + 1416 * q^91 + 3744 * q^92 + 96 * q^93 - 2032 * 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 $

$2$	−1.61559	−0.571198	−0.285599	−	0.958349i	$-0.592192\pi$
$2$	−1.61559	−0.571198	−0.285599	+	0.958349i	$0.592192\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−5.38987	−0.673733
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	4.84677	0.329781
$7$	−11.6241	−0.627641	−0.313820	−	0.949482i	$-0.601609\pi$
$7$	−11.6241	−0.627641	−0.313820	+	0.949482i	$0.601609\pi$
$8$	21.6325	0.956032
$9$	9.00000	0.333333
$10$	−8.07795	−0.255447
$11$	0	0
$11$	0	0
$12$	16.1696	0.388980
$13$	−54.8892	−1.17104	−0.585520	−	0.810658i	$-0.699110\pi$
$13$	−54.8892	−1.17104	−0.585520	+	0.810658i	$0.699110\pi$
$14$	18.7797	0.358507
$15$	−15.0000	−0.258199
$16$	8.16960	0.127650
$17$	81.3685	1.16087	0.580434	−	0.814307i	$-0.302883\pi$
$17$	81.3685	1.16087	0.580434	+	0.814307i	$0.302883\pi$
$18$	−14.5403	−0.190399
$19$	22.6183	0.273105	0.136552	−	0.990633i	$-0.456398\pi$
$19$	22.6183	0.273105	0.136552	+	0.990633i	$0.456398\pi$
$20$	−26.9493	−0.301303
$21$	34.8722	0.362368
$22$	0	0
$23$	−150.238	−1.36203	−0.681017	−	0.732268i	$-0.738462\pi$
$23$	−150.238	−1.36203	−0.681017	+	0.732268i	$0.738462\pi$
$24$	−64.8976	−0.551966
$25$	25.0000	0.200000
$26$	88.6784	0.668895
$27$	−27.0000	−0.192450
$28$	62.6522	0.422862
$29$	−18.7572	−0.120108	−0.0600539	−	0.998195i	$-0.519127\pi$
$29$	−18.7572	−0.120108	−0.0600539	+	0.998195i	$0.519127\pi$
$30$	24.2339	0.147483
$31$	−8.00000	−0.0463498	−0.0231749	−	0.999731i	$-0.507377\pi$
$31$	−8.00000	−0.0463498	−0.0231749	+	0.999731i	$0.507377\pi$
$32$	−186.259	−1.02895
$33$	0	0
$34$	−131.458	−0.663085
$35$	−58.1203	−0.280689
$36$	−48.5088	−0.224578
$37$	113.797	0.505626	0.252813	−	0.967515i	$-0.418644\pi$
$37$	113.797	0.505626	0.252813	+	0.967515i	$0.418644\pi$
$38$	−36.5419	−0.155997
$39$	164.667	0.676100
$40$	108.163	0.427551
$41$	−208.521	−0.794283	−0.397141	−	0.917757i	$-0.629998\pi$
$41$	−208.521	−0.794283	−0.397141	+	0.917757i	$0.629998\pi$
$42$	−56.3392	−0.206984
$43$	182.083	0.645754	0.322877	−	0.946441i	$-0.395350\pi$
$43$	182.083	0.645754	0.322877	+	0.946441i	$0.395350\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	242.723	0.777990
$47$	413.154	1.28223	0.641114	−	0.767446i	$-0.278473\pi$
$47$	413.154	1.28223	0.641114	+	0.767446i	$0.278473\pi$
$48$	−24.5088	−0.0736988
$49$	−207.881	−0.606067
$50$	−40.3898	−0.114240
$51$	−244.105	−0.670227
$52$	295.845	0.788968
$53$	222.476	0.576592	0.288296	−	0.957541i	$-0.406911\pi$
$53$	222.476	0.576592	0.288296	+	0.957541i	$0.406911\pi$
$54$	43.6209	0.109927
$55$	0	0
$56$	−251.458	−0.600045
$57$	−67.8548	−0.157677
$58$	30.3040	0.0686053
$59$	55.7973	0.123122	0.0615610	−	0.998103i	$-0.480392\pi$
$59$	55.7973	0.123122	0.0615610	+	0.998103i	$0.480392\pi$
$60$	80.8480	0.173957
$61$	233.823	0.490786	0.245393	−	0.969424i	$-0.421083\pi$
$61$	233.823	0.490786	0.245393	+	0.969424i	$0.421083\pi$
$62$	12.9247	0.0264749
$63$	−104.617	−0.209214
$64$	235.562	0.460081
$65$	−274.446	−0.523705
$66$	0	0
$67$	274.238	0.500052	0.250026	−	0.968239i	$-0.419561\pi$
$67$	274.238	0.500052	0.250026	+	0.968239i	$0.419561\pi$
$68$	−438.565	−0.782115
$69$	450.714	0.786370
$70$	93.8987	0.160329
$71$	−39.7269	−0.0664045	−0.0332022	−	0.999449i	$-0.510571\pi$
$71$	−39.7269	−0.0664045	−0.0332022	+	0.999449i	$0.510571\pi$
$72$	194.693	0.318677
$73$	623.250	0.999258	0.499629	−	0.866239i	$-0.333470\pi$
$73$	623.250	0.999258	0.499629	+	0.866239i	$0.333470\pi$
$74$	−183.850	−0.288812
$75$	−75.0000	−0.115470
$76$	−121.909	−0.184000
$77$	0	0
$78$	−266.035	−0.386187
$79$	152.798	0.217609	0.108804	−	0.994063i	$-0.465298\pi$
$79$	152.798	0.217609	0.108804	+	0.994063i	$0.465298\pi$
$80$	40.8480	0.0570868
$81$	81.0000	0.111111
$82$	336.885	0.453692
$83$	−992.373	−1.31237	−0.656187	−	0.754598i	$-0.727832\pi$
$83$	−992.373	−1.31237	−0.656187	+	0.754598i	$0.727832\pi$
$84$	−187.957	−0.244140
$85$	406.842	0.519156
$86$	−294.172	−0.368853
$87$	56.2717	0.0693443
$88$	0	0
$89$	−407.154	−0.484924	−0.242462	−	0.970161i	$-0.577955\pi$
$89$	−407.154	−0.484924	−0.242462	+	0.970161i	$0.577955\pi$
$90$	−72.7016	−0.0851491
$91$	638.035	0.734992
$92$	809.762	0.917647
$93$	24.0000	0.0267600
$94$	−667.488	−0.732406
$95$	113.091	0.122136
$96$	558.777	0.594062
$97$	−602.678	−0.630853	−0.315426	−	0.948950i	$-0.602148\pi$
$97$	−602.678	−0.630853	−0.315426	+	0.948950i	$0.602148\pi$
$98$	335.851	0.346184
$99$	0	0
$100$	−134.747	−0.134747
$101$	−870.120	−0.857230	−0.428615	−	0.903487i	$-0.640998\pi$
$101$	−870.120	−0.857230	−0.428615	+	0.903487i	$0.640998\pi$
$102$	394.374	0.382832
$103$	1299.67	1.24330	0.621650	−	0.783295i	$-0.286463\pi$
$103$	1299.67	1.24330	0.621650	+	0.783295i	$0.286463\pi$
$104$	−1187.39	−1.11955
$105$	174.361	0.162056
$106$	−359.430	−0.329348
$107$	1665.64	1.50489	0.752445	−	0.658655i	$-0.228875\pi$
$107$	1665.64	1.50489	0.752445	+	0.658655i	$0.228875\pi$
$108$	145.526	0.129660
$109$	2256.41	1.98280	0.991399	−	0.130876i	$-0.0417788\pi$
$109$	2256.41	1.98280	0.991399	+	0.130876i	$0.0417788\pi$
$110$	0	0
$111$	−341.392	−0.291923
$112$	−94.9640	−0.0801183
$113$	−581.868	−0.484403	−0.242201	−	0.970226i	$-0.577869\pi$
$113$	−581.868	−0.484403	−0.242201	+	0.970226i	$0.577869\pi$
$114$	109.626	0.0900647
$115$	−751.189	−0.609120
$116$	101.099	0.0809207
$117$	−494.002	−0.390346
$118$	−90.1457	−0.0703270
$119$	−945.833	−0.728608
$120$	−324.488	−0.246847
$121$	0	0
$122$	−377.762	−0.280336
$123$	625.564	0.458579
$124$	43.1189	0.0312274
$125$	125.000	0.0894427
$126$	169.018	0.119502
$127$	2041.08	1.42612	0.713059	−	0.701105i	$-0.247309\pi$
$127$	2041.08	1.42612	0.713059	+	0.701105i	$0.247309\pi$
$128$	1109.50	0.766148
$129$	−546.249	−0.372826
$130$	443.392	0.299139
$131$	2062.95	1.37588	0.687942	−	0.725766i	$-0.258515\pi$
$131$	2062.95	1.37588	0.687942	+	0.725766i	$0.258515\pi$
$132$	0	0
$133$	−262.916	−0.171412
$134$	−443.056	−0.285629
$135$	−135.000	−0.0860663
$136$	1760.21	1.10983
$137$	−1121.39	−0.699321	−0.349661	−	0.936876i	$-0.613703\pi$
$137$	−1121.39	−0.699321	−0.349661	+	0.936876i	$0.613703\pi$
$138$	−728.169	−0.449173
$139$	375.119	0.228901	0.114450	−	0.993429i	$-0.463489\pi$
$139$	375.119	0.228901	0.114450	+	0.993429i	$0.463489\pi$
$140$	313.261	0.189110
$141$	−1239.46	−0.740295
$142$	64.1824	0.0379301
$143$	0	0
$144$	73.5264	0.0425500
$145$	−93.7861	−0.0537139
$146$	−1006.92	−0.570774
$147$	623.643	0.349913
$148$	−613.353	−0.340657
$149$	516.114	0.283770	0.141885	−	0.989883i	$-0.454684\pi$
$149$	516.114	0.283770	0.141885	+	0.989883i	$0.454684\pi$
$150$	121.169	0.0659562
$151$	−1205.51	−0.649688	−0.324844	−	0.945768i	$-0.605312\pi$
$151$	−1205.51	−0.649688	−0.324844	+	0.945768i	$0.605312\pi$
$152$	489.291	0.261097
$153$	732.316	0.386956
$154$	0	0
$155$	−40.0000	−0.0207282
$156$	−887.536	−0.455511
$157$	−3583.23	−1.82148	−0.910742	−	0.412975i	$-0.864490\pi$
$157$	−3583.23	−1.82148	−0.910742	+	0.412975i	$0.864490\pi$
$158$	−246.859	−0.124298
$159$	−667.427	−0.332896
$160$	−931.295	−0.460159
$161$	1746.38	0.854867
$162$	−130.863	−0.0634664
$163$	−2820.14	−1.35516	−0.677578	−	0.735451i	$-0.736970\pi$
$163$	−2820.14	−1.35516	−0.677578	+	0.735451i	$0.736970\pi$
$164$	1123.90	0.535135
$165$	0	0
$166$	1603.27	0.749625
$167$	−1254.40	−0.581248	−0.290624	−	0.956837i	$-0.593863\pi$
$167$	−1254.40	−0.581248	−0.290624	+	0.956837i	$0.593863\pi$
$168$	754.374	0.346436
$169$	815.819	0.371333
$170$	−657.291	−0.296541
$171$	203.564	0.0910349
$172$	−981.404	−0.435066
$173$	2247.03	0.987505	0.493753	−	0.869602i	$-0.335625\pi$
$173$	2247.03	0.987505	0.493753	+	0.869602i	$0.335625\pi$
$174$	−90.9120	−0.0396093
$175$	−290.602	−0.125528
$176$	0	0
$177$	−167.392	−0.0710845
$178$	657.794	0.276988
$179$	1479.20	0.617656	0.308828	−	0.951118i	$-0.400063\pi$
$179$	1479.20	0.617656	0.308828	+	0.951118i	$0.400063\pi$
$180$	−242.544	−0.100434
$181$	−2722.34	−1.11796	−0.558978	−	0.829182i	$-0.688807\pi$
$181$	−2722.34	−1.11796	−0.558978	+	0.829182i	$0.688807\pi$
$182$	−1030.80	−0.419826
$183$	−701.469	−0.283356
$184$	−3250.03	−1.30215
$185$	568.987	0.226123
$186$	−38.7742	−0.0152853
$187$	0	0
$188$	−2226.85	−0.863880
$189$	313.850	0.120789
$190$	−182.709	−0.0697638
$191$	327.383	0.124024	0.0620121	−	0.998075i	$-0.480248\pi$
$191$	327.383	0.124024	0.0620121	+	0.998075i	$0.480248\pi$
$192$	−706.685	−0.265628
$193$	4014.03	1.49708	0.748539	−	0.663091i	$-0.230756\pi$
$193$	4014.03	1.49708	0.748539	+	0.663091i	$0.230756\pi$
$194$	973.682	0.360342
$195$	823.337	0.302361
$196$	1120.45	0.408328
$197$	−3748.96	−1.35585	−0.677925	−	0.735131i	$-0.737121\pi$
$197$	−3748.96	−1.35585	−0.677925	+	0.735131i	$0.737121\pi$
$198$	0	0
$199$	3348.70	1.19288	0.596439	−	0.802658i	$-0.296582\pi$
$199$	3348.70	1.19288	0.596439	+	0.802658i	$0.296582\pi$
$200$	540.814	0.191206
$201$	−822.714	−0.288705
$202$	1405.76	0.489647
$203$	218.035	0.0753846
$204$	1315.70	0.451555
$205$	−1042.61	−0.355214
$206$	−2099.73	−0.710169
$207$	−1352.14	−0.454011
$208$	−448.422	−0.149483
$209$	0	0
$210$	−281.696	−0.0925661
$211$	−3657.26	−1.19325	−0.596626	−	0.802519i	$-0.703493\pi$
$211$	−3657.26	−1.19325	−0.596626	+	0.802519i	$0.703493\pi$
$212$	−1199.11	−0.388469
$213$	119.181	0.0383386
$214$	−2690.99	−0.859589
$215$	910.415	0.288790
$216$	−584.079	−0.183989
$217$	92.9925	0.0290910
$218$	−3645.44	−1.13257
$219$	−1869.75	−0.576922
$220$	0	0
$221$	−4466.25	−1.35942
$222$	551.550	0.166746
$223$	−4717.04	−1.41649	−0.708243	−	0.705969i	$-0.750512\pi$
$223$	−4717.04	−1.41649	−0.708243	+	0.705969i	$0.750512\pi$
$224$	2165.09	0.645808
$225$	225.000	0.0666667
$226$	940.060	0.276690
$227$	−5656.43	−1.65388	−0.826939	−	0.562291i	$-0.809920\pi$
$227$	−5656.43	−1.65388	−0.826939	+	0.562291i	$0.809920\pi$
$228$	365.728	0.106232
$229$	−5876.85	−1.69586	−0.847932	−	0.530105i	$-0.822153\pi$
$229$	−5876.85	−1.69586	−0.847932	+	0.530105i	$0.822153\pi$
$230$	1213.61	0.347928
$231$	0	0
$232$	−405.766	−0.114827
$233$	4199.46	1.18076	0.590378	−	0.807127i	$-0.298979\pi$
$233$	4199.46	1.18076	0.590378	+	0.807127i	$0.298979\pi$
$234$	798.106	0.222965
$235$	2065.77	0.573430
$236$	−300.740	−0.0829514
$237$	−458.393	−0.125637
$238$	1528.08	0.416179
$239$	3791.25	1.02609	0.513045	−	0.858362i	$-0.328518\pi$
$239$	3791.25	1.02609	0.513045	+	0.858362i	$0.328518\pi$
$240$	−122.544	−0.0329591
$241$	−5476.07	−1.46367	−0.731836	−	0.681481i	$-0.761336\pi$
$241$	−5476.07	−1.46367	−0.731836	+	0.681481i	$0.761336\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	−1260.27	−0.330659
$245$	−1039.41	−0.271042
$246$	−1010.66	−0.261939
$247$	−1241.50	−0.319816
$248$	−173.060	−0.0443119
$249$	2977.12	0.757700
$250$	−201.949	−0.0510895
$251$	−5038.04	−1.26693	−0.633463	−	0.773773i	$-0.718367\pi$
$251$	−5038.04	−1.26693	−0.633463	+	0.773773i	$0.718367\pi$
$252$	563.870	0.140954
$253$	0	0
$254$	−3297.56	−0.814595
$255$	−1220.53	−0.299735
$256$	−3676.99	−0.897703
$257$	2976.85	0.722531	0.361266	−	0.932463i	$-0.382345\pi$
$257$	2976.85	0.722531	0.361266	+	0.932463i	$0.382345\pi$
$258$	882.515	0.212957
$259$	−1322.79	−0.317352
$260$	1479.23	0.352837
$261$	−168.815	−0.0400360
$262$	−3332.88	−0.785901
$263$	1219.36	0.285889	0.142944	−	0.989731i	$-0.454343\pi$
$263$	1219.36	0.285889	0.142944	+	0.989731i	$0.454343\pi$
$264$	0	0
$265$	1112.38	0.257860
$266$	424.765	0.0979098
$267$	1221.46	0.279971
$268$	−1478.11	−0.336902
$269$	−5035.93	−1.14143	−0.570717	−	0.821147i	$-0.693335\pi$
$269$	−5035.93	−1.14143	−0.570717	+	0.821147i	$0.693335\pi$
$270$	218.105	0.0491609
$271$	−1308.96	−0.293407	−0.146704	−	0.989180i	$-0.546866\pi$
$271$	−1308.96	−0.293407	−0.146704	+	0.989180i	$0.546866\pi$
$272$	664.748	0.148185
$273$	−1914.11	−0.424348
$274$	1811.71	0.399450
$275$	0	0
$276$	−2429.29	−0.529804
$277$	−6094.53	−1.32197	−0.660983	−	0.750401i	$-0.729861\pi$
$277$	−6094.53	−1.32197	−0.660983	+	0.750401i	$0.729861\pi$
$278$	−606.040	−0.130748
$279$	−72.0000	−0.0154499
$280$	−1257.29	−0.268348
$281$	−5075.12	−1.07742	−0.538712	−	0.842490i	$-0.681089\pi$
$281$	−5075.12	−1.07742	−0.538712	+	0.842490i	$0.681089\pi$
$282$	2002.46	0.422855
$283$	5963.23	1.25257	0.626285	−	0.779594i	$-0.284575\pi$
$283$	5963.23	1.25257	0.626285	+	0.779594i	$0.284575\pi$
$284$	214.123	0.0447389
$285$	−339.274	−0.0705153
$286$	0	0
$287$	2423.87	0.498524
$288$	−1676.33	−0.342982
$289$	1707.83	0.347614
$290$	151.520	0.0306812
$291$	1808.04	0.364223
$292$	−3359.23	−0.673234
$293$	1422.61	0.283651	0.141826	−	0.989892i	$-0.454703\pi$
$293$	1422.61	0.283651	0.141826	+	0.989892i	$0.454703\pi$
$294$	−1007.55	−0.199869
$295$	278.987	0.0550618
$296$	2461.73	0.483395
$297$	0	0
$298$	−833.828	−0.162089
$299$	8246.43	1.59499
$300$	404.240	0.0777960
$301$	−2116.55	−0.405301
$302$	1947.61	0.371100
$303$	2610.36	0.494922
$304$	184.782	0.0348618
$305$	1169.11	0.219486
$306$	−1183.12	−0.221028
$307$	7796.85	1.44948	0.724739	−	0.689023i	$-0.241960\pi$
$307$	7796.85	1.44948	0.724739	+	0.689023i	$0.241960\pi$
$308$	0	0
$309$	−3899.00	−0.717819
$310$	64.6236	0.0118399
$311$	−3214.52	−0.586105	−0.293053	−	0.956096i	$-0.594671\pi$
$311$	−3214.52	−0.586105	−0.293053	+	0.956096i	$0.594671\pi$
$312$	3562.18	0.646373
$313$	−135.877	−0.0245375	−0.0122687	−	0.999925i	$-0.503905\pi$
$313$	−135.877	−0.0245375	−0.0122687	+	0.999925i	$0.503905\pi$
$314$	5789.04	1.04043
$315$	−523.083	−0.0935631
$316$	−823.560	−0.146610
$317$	−6856.78	−1.21487	−0.607437	−	0.794367i	$-0.707802\pi$
$317$	−6856.78	−1.21487	−0.607437	+	0.794367i	$0.707802\pi$
$318$	1078.29	0.190149
$319$	0	0
$320$	1177.81	0.205755
$321$	−4996.91	−0.868848
$322$	−2821.43	−0.488298
$323$	1840.41	0.317038
$324$	−436.579	−0.0748593
$325$	−1372.23	−0.234208
$326$	4556.19	0.774062
$327$	−6769.23	−1.14477
$328$	−4510.85	−0.759360
$329$	−4802.53	−0.804779
$330$	0	0
$331$	−5976.77	−0.992487	−0.496244	−	0.868183i	$-0.665288\pi$
$331$	−5976.77	−0.992487	−0.496244	+	0.868183i	$0.665288\pi$
$332$	5348.76	0.884190
$333$	1024.18	0.168542
$334$	2026.60	0.332008
$335$	1371.19	0.223630
$336$	284.892	0.0462563
$337$	174.286	0.0281720	0.0140860	−	0.999901i	$-0.495516\pi$
$337$	174.286	0.0281720	0.0140860	+	0.999901i	$0.495516\pi$
$338$	−1318.03	−0.212105
$339$	1745.60	0.279670
$340$	−2192.83	−0.349773
$341$	0	0
$342$	−328.877	−0.0519989
$343$	6403.48	1.00803
$344$	3938.92	0.617362
$345$	2253.57	0.351675
$346$	−3630.28	−0.564061
$347$	−7455.62	−1.15343	−0.576713	−	0.816947i	$-0.695665\pi$
$347$	−7455.62	−1.15343	−0.576713	+	0.816947i	$0.695665\pi$
$348$	−303.297	−0.0467196
$349$	852.736	0.130791	0.0653953	−	0.997859i	$-0.479169\pi$
$349$	852.736	0.130791	0.0653953	+	0.997859i	$0.479169\pi$
$350$	469.493	0.0717014
$351$	1482.01	0.225367
$352$	0	0
$353$	−5002.15	−0.754214	−0.377107	−	0.926170i	$-0.623081\pi$
$353$	−5002.15	−0.754214	−0.377107	+	0.926170i	$0.623081\pi$
$354$	270.437	0.0406033
$355$	−198.635	−0.0296970
$356$	2194.51	0.326710
$357$	2837.50	0.420662
$358$	−2389.78	−0.352804
$359$	−5362.16	−0.788311	−0.394156	−	0.919044i	$-0.628963\pi$
$359$	−5362.16	−0.788311	−0.394156	+	0.919044i	$0.628963\pi$
$360$	973.464	0.142517
$361$	−6347.41	−0.925414
$362$	4398.19	0.638574
$363$	0	0
$364$	−3438.92	−0.495189
$365$	3116.25	0.446882
$366$	1133.29	0.161852
$367$	7357.18	1.04644	0.523218	−	0.852199i	$-0.324732\pi$
$367$	7357.18	1.04644	0.523218	+	0.852199i	$0.324732\pi$
$368$	−1227.38	−0.173864
$369$	−1876.69	−0.264761
$370$	−919.250	−0.129161
$371$	−2586.07	−0.361893
$372$	−129.357	−0.0180291
$373$	−12775.1	−1.77337	−0.886685	−	0.462373i	$-0.846998\pi$
$373$	−12775.1	−1.77337	−0.886685	+	0.462373i	$0.846998\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	8937.58	1.22585
$377$	1029.57	0.140651
$378$	−507.053	−0.0689947
$379$	−4988.63	−0.676117	−0.338059	−	0.941125i	$-0.609770\pi$
$379$	−4988.63	−0.676117	−0.338059	+	0.941125i	$0.609770\pi$
$380$	−609.547	−0.0822871
$381$	−6123.25	−0.823369
$382$	−528.918	−0.0708424
$383$	−11237.9	−1.49929	−0.749645	−	0.661840i	$-0.769776\pi$
$383$	−11237.9	−1.49929	−0.749645	+	0.661840i	$0.769776\pi$
$384$	−3328.50	−0.442336
$385$	0	0
$386$	−6485.02	−0.855127
$387$	1638.75	0.215251
$388$	3248.36	0.425027
$389$	4888.55	0.637170	0.318585	−	0.947894i	$-0.396792\pi$
$389$	4888.55	0.637170	0.318585	+	0.947894i	$0.396792\pi$
$390$	−1330.18	−0.172708
$391$	−12224.6	−1.58114
$392$	−4497.00	−0.579420
$393$	−6188.85	−0.794366
$394$	6056.79	0.774458
$395$	763.989	0.0973176
$396$	0	0
$397$	9876.18	1.24854	0.624271	−	0.781208i	$-0.285396\pi$
$397$	9876.18	1.24854	0.624271	+	0.781208i	$0.285396\pi$
$398$	−5410.12	−0.681369
$399$	788.749	0.0989645
$400$	204.240	0.0255300
$401$	−1653.99	−0.205976	−0.102988	−	0.994683i	$-0.532840\pi$
$401$	−1653.99	−0.205976	−0.102988	+	0.994683i	$0.532840\pi$
$402$	1329.17	0.164908
$403$	439.113	0.0542774
$404$	4689.83	0.577544
$405$	405.000	0.0496904
$406$	−352.256	−0.0430595
$407$	0	0
$408$	−5280.62	−0.640759
$409$	−7547.30	−0.912445	−0.456222	−	0.889866i	$-0.650798\pi$
$409$	−7547.30	−0.912445	−0.456222	+	0.889866i	$0.650798\pi$
$410$	1684.43	0.202897
$411$	3364.18	0.403753
$412$	−7005.02	−0.837652
$413$	−648.592	−0.0772763
$414$	2184.51	0.259330
$415$	−4961.86	−0.586912
$416$	10223.6	1.20494
$417$	−1125.36	−0.132156
$418$	0	0
$419$	−6605.31	−0.770145	−0.385072	−	0.922886i	$-0.625824\pi$
$419$	−6605.31	−0.770145	−0.385072	+	0.922886i	$0.625824\pi$
$420$	−939.783	−0.109183
$421$	−12427.1	−1.43862	−0.719311	−	0.694688i	$-0.755542\pi$
$421$	−12427.1	−1.43862	−0.719311	+	0.694688i	$0.755542\pi$
$422$	5908.64	0.681583
$423$	3718.39	0.427409
$424$	4812.72	0.551241
$425$	2034.21	0.232174
$426$	−192.547	−0.0218989
$427$	−2717.97	−0.308037
$428$	−8977.56	−1.01389
$429$	0	0
$430$	−1470.86	−0.164956
$431$	3377.95	0.377518	0.188759	−	0.982023i	$-0.439554\pi$
$431$	3377.95	0.377518	0.188759	+	0.982023i	$0.439554\pi$
$432$	−220.579	−0.0245663
$433$	−6972.88	−0.773892	−0.386946	−	0.922102i	$-0.626470\pi$
$433$	−6972.88	−0.773892	−0.386946	+	0.922102i	$0.626470\pi$
$434$	−150.238	−0.0166167
$435$	281.358	0.0310117
$436$	−12161.8	−1.33588
$437$	−3398.12	−0.371977
$438$	3020.75	0.329536
$439$	−5450.66	−0.592587	−0.296293	−	0.955097i	$-0.595751\pi$
$439$	−5450.66	−0.592587	−0.296293	+	0.955097i	$0.595751\pi$
$440$	0	0
$441$	−1870.93	−0.202022
$442$	7215.63	0.776498
$443$	−1441.45	−0.154595	−0.0772974	−	0.997008i	$-0.524629\pi$
$443$	−1441.45	−0.154595	−0.0772974	+	0.997008i	$0.524629\pi$
$444$	1840.06	0.196679
$445$	−2035.77	−0.216865
$446$	7620.80	0.809093
$447$	−1548.34	−0.163834
$448$	−2738.18	−0.288766
$449$	−6059.89	−0.636936	−0.318468	−	0.947934i	$-0.603168\pi$
$449$	−6059.89	−0.636936	−0.318468	+	0.947934i	$0.603168\pi$
$450$	−363.508	−0.0380798
$451$	0	0
$452$	3136.19	0.326358
$453$	3616.52	0.375097
$454$	9138.47	0.944691
$455$	3190.18	0.328698
$456$	−1467.87	−0.150744
$457$	8753.62	0.896011	0.448005	−	0.894031i	$-0.352135\pi$
$457$	8753.62	0.896011	0.448005	+	0.894031i	$0.352135\pi$
$458$	9494.58	0.968673
$459$	−2196.95	−0.223409
$460$	4048.81	0.410384
$461$	13507.0	1.36461	0.682305	−	0.731068i	$-0.260978\pi$
$461$	13507.0	1.36461	0.682305	+	0.731068i	$0.260978\pi$
$462$	0	0
$463$	7710.78	0.773975	0.386988	−	0.922085i	$-0.373516\pi$
$463$	7710.78	0.773975	0.386988	+	0.922085i	$0.373516\pi$
$464$	−153.239	−0.0153318
$465$	120.000	0.0119675
$466$	−6784.61	−0.674444
$467$	19571.0	1.93926	0.969632	−	0.244568i	$-0.0786462\pi$
$467$	19571.0	1.93926	0.969632	+	0.244568i	$0.0786462\pi$
$468$	2662.61	0.262989
$469$	−3187.76	−0.313853
$470$	−3337.44	−0.327542
$471$	10749.7	1.05163
$472$	1207.04	0.117709
$473$	0	0
$474$	740.576	0.0717633
$475$	565.457	0.0546209
$476$	5097.91	0.490887
$477$	2002.28	0.192197
$478$	−6125.10	−0.586100
$479$	−17127.1	−1.63373	−0.816864	−	0.576830i	$-0.804289\pi$
$479$	−17127.1	−1.63373	−0.816864	+	0.576830i	$0.804289\pi$
$480$	2793.89	0.265673
$481$	−6246.24	−0.592108
$482$	8847.09	0.836046
$483$	−5239.13	−0.493558
$484$	0	0
$485$	−3013.39	−0.282126
$486$	392.589	0.0366423
$487$	−3404.05	−0.316740	−0.158370	−	0.987380i	$-0.550624\pi$
$487$	−3404.05	−0.316740	−0.158370	+	0.987380i	$0.550624\pi$
$488$	5058.18	0.469207
$489$	8460.42	0.782400
$490$	1679.25	0.154818
$491$	4745.15	0.436142	0.218071	−	0.975933i	$-0.430024\pi$
$491$	4745.15	0.436142	0.218071	+	0.975933i	$0.430024\pi$
$492$	−3371.71	−0.308960
$493$	−1526.25	−0.139429
$494$	2005.75	0.182678
$495$	0	0
$496$	−65.3568	−0.00591655
$497$	461.788	0.0416781
$498$	−4809.81	−0.432796
$499$	−11514.8	−1.03301	−0.516504	−	0.856285i	$-0.672767\pi$
$499$	−11514.8	−1.03301	−0.516504	+	0.856285i	$0.672767\pi$
$500$	−673.733	−0.0602605
$501$	3763.20	0.335584
$502$	8139.42	0.723665
$503$	3538.22	0.313641	0.156820	−	0.987627i	$-0.449876\pi$
$503$	3538.22	0.313641	0.156820	+	0.987627i	$0.449876\pi$
$504$	−2263.12	−0.200015
$505$	−4350.60	−0.383365
$506$	0	0
$507$	−2447.46	−0.214389
$508$	−11001.2	−0.960823
$509$	16969.4	1.47771	0.738857	−	0.673862i	$-0.235366\pi$
$509$	16969.4	1.47771	0.738857	+	0.673862i	$0.235366\pi$
$510$	1971.87	0.171208
$511$	−7244.70	−0.627175
$512$	−2935.50	−0.253382
$513$	−610.693	−0.0525590
$514$	−4809.36	−0.412708
$515$	6498.33	0.556020
$516$	2944.21	0.251185
$517$	0	0
$518$	2137.08	0.181270
$519$	−6741.09	−0.570136
$520$	−5936.96	−0.500679
$521$	7146.45	0.600944	0.300472	−	0.953791i	$-0.402856\pi$
$521$	7146.45	0.600944	0.300472	+	0.953791i	$0.402856\pi$
$522$	272.736	0.0228684
$523$	18268.7	1.52741	0.763704	−	0.645566i	$-0.223378\pi$
$523$	18268.7	1.52741	0.763704	+	0.645566i	$0.223378\pi$
$524$	−11119.0	−0.926978
$525$	871.805	0.0724737
$526$	−1969.98	−0.163299
$527$	−650.948	−0.0538059
$528$	0	0
$529$	10404.4	0.855134
$530$	−1797.15	−0.147289
$531$	502.176	0.0410406
$532$	1417.08	0.115486
$533$	11445.6	0.930136
$534$	−1973.38	−0.159919
$535$	8328.18	0.673007
$536$	5932.46	0.478066
$537$	−4437.59	−0.356604
$538$	8136.00	0.651985
$539$	0	0
$540$	727.632	0.0579857
$541$	6790.66	0.539655	0.269827	−	0.962909i	$-0.413033\pi$
$541$	6790.66	0.539655	0.269827	+	0.962909i	$0.413033\pi$
$542$	2114.74	0.167594
$543$	8167.03	0.645453
$544$	−15155.6	−1.19447
$545$	11282.1	0.886734
$546$	3092.41	0.242386
$547$	−1730.48	−0.135265	−0.0676326	−	0.997710i	$-0.521545\pi$
$547$	−1730.48	−0.135265	−0.0676326	+	0.997710i	$0.521545\pi$
$548$	6044.15	0.471156
$549$	2104.41	0.163595
$550$	0	0
$551$	−424.256	−0.0328020
$552$	9750.08	0.751795
$553$	−1776.13	−0.136580
$554$	9846.26	0.755104
$555$	−1706.96	−0.130552
$556$	−2021.84	−0.154218
$557$	21036.7	1.60028	0.800139	−	0.599814i	$-0.204759\pi$
$557$	21036.7	1.60028	0.800139	+	0.599814i	$0.204759\pi$
$558$	116.323	0.00882496
$559$	−9994.39	−0.756203
$560$	−474.820	−0.0358300
$561$	0	0
$562$	8199.32	0.615422
$563$	−8817.80	−0.660082	−0.330041	−	0.943967i	$-0.607062\pi$
$563$	−8817.80	−0.660082	−0.330041	+	0.943967i	$0.607062\pi$
$564$	6680.54	0.498761
$565$	−2909.34	−0.216632
$566$	−9634.14	−0.715465
$567$	−941.549	−0.0697378
$568$	−859.394	−0.0634848
$569$	4558.26	0.335839	0.167919	−	0.985801i	$-0.446295\pi$
$569$	4558.26	0.335839	0.167919	+	0.985801i	$0.446295\pi$
$570$	548.128	0.0402782
$571$	−11459.8	−0.839892	−0.419946	−	0.907549i	$-0.637951\pi$
$571$	−11459.8	−0.839892	−0.419946	+	0.907549i	$0.637951\pi$
$572$	0	0
$573$	−982.150	−0.0716054
$574$	−3915.98	−0.284756
$575$	−3755.95	−0.272407
$576$	2120.05	0.153360
$577$	−14278.4	−1.03019	−0.515093	−	0.857134i	$-0.672243\pi$
$577$	−14278.4	−1.03019	−0.515093	+	0.857134i	$0.672243\pi$
$578$	−2759.15	−0.198556
$579$	−12042.1	−0.864338
$580$	505.495	0.0361888
$581$	11535.4	0.823700
$582$	−2921.04	−0.208043
$583$	0	0
$584$	13482.5	0.955323
$585$	−2470.01	−0.174568
$586$	−2298.36	−0.162021
$587$	−2726.88	−0.191738	−0.0958692	−	0.995394i	$-0.530563\pi$
$587$	−2726.88	−0.191738	−0.0958692	+	0.995394i	$0.530563\pi$
$588$	−3361.35	−0.235748
$589$	−180.946	−0.0126583
$590$	−450.728	−0.0314512
$591$	11246.9	0.782800
$592$	929.679	0.0645432
$593$	−22458.5	−1.55525	−0.777623	−	0.628731i	$-0.783575\pi$
$593$	−22458.5	−1.55525	−0.777623	+	0.628731i	$0.783575\pi$
$594$	0	0
$595$	−4729.16	−0.325843
$596$	−2781.78	−0.191185
$597$	−10046.1	−0.688708
$598$	−13322.9	−0.911057
$599$	9565.45	0.652477	0.326239	−	0.945287i	$-0.394219\pi$
$599$	9565.45	0.652477	0.326239	+	0.945287i	$0.394219\pi$
$600$	−1622.44	−0.110393
$601$	−17480.9	−1.18646	−0.593229	−	0.805034i	$-0.702147\pi$
$601$	−17480.9	−1.18646	−0.593229	+	0.805034i	$0.702147\pi$
$602$	3419.47	0.231507
$603$	2468.14	0.166684
$604$	6497.53	0.437716
$605$	0	0
$606$	−4217.27	−0.282698
$607$	−7721.80	−0.516339	−0.258170	−	0.966100i	$-0.583119\pi$
$607$	−7721.80	−0.516339	−0.258170	+	0.966100i	$0.583119\pi$
$608$	−4212.86	−0.281010
$609$	−654.106	−0.0435233
$610$	−1888.81	−0.125370
$611$	−22677.7	−1.50154
$612$	−3947.09	−0.260705
$613$	4520.81	0.297869	0.148935	−	0.988847i	$-0.452416\pi$
$613$	4520.81	0.297869	0.148935	+	0.988847i	$0.452416\pi$
$614$	−12596.5	−0.827938
$615$	3127.82	0.205083
$616$	0	0
$617$	23275.9	1.51872	0.759361	−	0.650670i	$-0.225512\pi$
$617$	23275.9	1.51872	0.759361	+	0.650670i	$0.225512\pi$
$618$	6299.18	0.410016
$619$	7343.70	0.476847	0.238423	−	0.971161i	$-0.423369\pi$
$619$	7343.70	0.476847	0.238423	+	0.971161i	$0.423369\pi$
$620$	215.595	0.0139653
$621$	4056.42	0.262123
$622$	5193.35	0.334782
$623$	4732.79	0.304358
$624$	1345.27	0.0863042
$625$	625.000	0.0400000
$626$	219.522	0.0140158
$627$	0	0
$628$	19313.1	1.22720
$629$	9259.52	0.586965
$630$	845.088	0.0534430
$631$	−24937.4	−1.57328	−0.786641	−	0.617411i	$-0.788182\pi$
$631$	−24937.4	−1.57328	−0.786641	+	0.617411i	$0.788182\pi$
$632$	3305.41	0.208041
$633$	10971.8	0.688925
$634$	11077.8	0.693934
$635$	10205.4	0.637779
$636$	3597.34	0.224283
$637$	11410.4	0.709729
$638$	0	0
$639$	−357.542	−0.0221348
$640$	5547.51	0.342632
$641$	−4957.00	−0.305444	−0.152722	−	0.988269i	$-0.548804\pi$
$641$	−4957.00	−0.305444	−0.152722	+	0.988269i	$0.548804\pi$
$642$	8072.96	0.496284
$643$	−8148.35	−0.499751	−0.249875	−	0.968278i	$-0.580390\pi$
$643$	−8148.35	−0.499751	−0.249875	+	0.968278i	$0.580390\pi$
$644$	−9412.73	−0.575953
$645$	−2731.25	−0.166733
$646$	−2973.36	−0.181091
$647$	13003.9	0.790162	0.395081	−	0.918646i	$-0.370716\pi$
$647$	13003.9	0.790162	0.395081	+	0.918646i	$0.370716\pi$
$648$	1752.24	0.106226
$649$	0	0
$650$	2216.96	0.133779
$651$	−278.978	−0.0167957
$652$	15200.2	0.913014
$653$	−10468.5	−0.627358	−0.313679	−	0.949529i	$-0.601562\pi$
$653$	−10468.5	−0.627358	−0.313679	+	0.949529i	$0.601562\pi$
$654$	10936.3	0.653889
$655$	10314.7	0.615314
$656$	−1703.54	−0.101390
$657$	5609.25	0.333086
$658$	7758.92	0.459688
$659$	1728.41	0.102169	0.0510845	−	0.998694i	$-0.483732\pi$
$659$	1728.41	0.102169	0.0510845	+	0.998694i	$0.483732\pi$
$660$	0	0
$661$	−4413.72	−0.259718	−0.129859	−	0.991532i	$-0.541453\pi$
$661$	−4413.72	−0.259718	−0.129859	+	0.991532i	$0.541453\pi$
$662$	9656.02	0.566906
$663$	13398.7	0.784863
$664$	−21467.6	−1.25467
$665$	−1314.58	−0.0766576
$666$	−1654.65	−0.0962708
$667$	2818.05	0.163591
$668$	6761.05	0.391606
$669$	14151.1	0.817808
$670$	−2215.28	−0.127737
$671$	0	0
$672$	−6495.26	−0.372858
$673$	−7859.70	−0.450177	−0.225089	−	0.974338i	$-0.572267\pi$
$673$	−7859.70	−0.450177	−0.225089	+	0.974338i	$0.572267\pi$
$674$	−281.575	−0.0160918
$675$	−675.000	−0.0384900
$676$	−4397.16	−0.250180
$677$	−11733.1	−0.666087	−0.333044	−	0.942911i	$-0.608076\pi$
$677$	−11733.1	−0.666087	−0.333044	+	0.942911i	$0.608076\pi$
$678$	−2820.18	−0.159747
$679$	7005.57	0.395949
$680$	8801.03	0.496330
$681$	16969.3	0.954867
$682$	0	0
$683$	−8357.23	−0.468200	−0.234100	−	0.972213i	$-0.575214\pi$
$683$	−8357.23	−0.468200	−0.234100	+	0.972213i	$0.575214\pi$
$684$	−1097.19	−0.0613332
$685$	−5606.96	−0.312746
$686$	−10345.4	−0.575786
$687$	17630.5	0.979108
$688$	1487.55	0.0824305
$689$	−12211.5	−0.675212
$690$	−3640.84	−0.200876
$691$	15588.4	0.858190	0.429095	−	0.903259i	$-0.358833\pi$
$691$	15588.4	0.858190	0.429095	+	0.903259i	$0.358833\pi$
$692$	−12111.2	−0.665315
$693$	0	0
$694$	12045.2	0.658834
$695$	1875.60	0.102368
$696$	1217.30	0.0662954
$697$	−16967.1	−0.922057
$698$	−1377.67	−0.0747073
$699$	−12598.4	−0.681709
$700$	1566.30	0.0845725
$701$	7896.13	0.425439	0.212719	−	0.977113i	$-0.431768\pi$
$701$	7896.13	0.425439	0.212719	+	0.977113i	$0.431768\pi$
$702$	−2394.32	−0.128729
$703$	2573.90	0.138089
$704$	0	0
$705$	−6197.31	−0.331070
$706$	8081.43	0.430805
$707$	10114.3	0.538032
$708$	902.221	0.0478920
$709$	9755.27	0.516737	0.258369	−	0.966046i	$-0.416815\pi$
$709$	9755.27	0.516737	0.258369	+	0.966046i	$0.416815\pi$
$710$	320.912	0.0169628
$711$	1375.18	0.0725363
$712$	−8807.78	−0.463603
$713$	1201.90	0.0631299
$714$	−4584.23	−0.240281
$715$	0	0
$716$	−7972.68	−0.416136
$717$	−11373.7	−0.592413
$718$	8663.05	0.450281
$719$	−15126.6	−0.784597	−0.392299	−	0.919838i	$-0.628320\pi$
$719$	−15126.6	−0.784597	−0.392299	+	0.919838i	$0.628320\pi$
$720$	367.632	0.0190289
$721$	−15107.4	−0.780345
$722$	10254.8	0.528594
$723$	16428.2	0.845051
$724$	14673.1	0.753205
$725$	−468.931	−0.0240216
$726$	0	0
$727$	38530.8	1.96565	0.982825	−	0.184542i	$-0.0590802\pi$
$727$	38530.8	1.96565	0.982825	+	0.184542i	$0.0590802\pi$
$728$	13802.3	0.702676
$729$	729.000	0.0370370
$730$	−5034.58	−0.255258
$731$	14815.8	0.749635
$732$	3780.82	0.190906
$733$	−32169.5	−1.62102	−0.810511	−	0.585723i	$-0.800811\pi$
$733$	−32169.5	−1.62102	−0.810511	+	0.585723i	$0.800811\pi$
$734$	−11886.2	−0.597721
$735$	3118.22	0.156486
$736$	27983.2	1.40146
$737$	0	0
$738$	3031.97	0.151231
$739$	10748.1	0.535015	0.267507	−	0.963556i	$-0.413800\pi$
$739$	10748.1	0.535015	0.267507	+	0.963556i	$0.413800\pi$
$740$	−3066.76	−0.152347
$741$	3724.49	0.184646
$742$	4178.04	0.206712
$743$	15232.5	0.752121	0.376060	−	0.926595i	$-0.377278\pi$
$743$	15232.5	0.752121	0.376060	+	0.926595i	$0.377278\pi$
$744$	519.181	0.0255835
$745$	2580.57	0.126906
$746$	20639.3	1.01295
$747$	−8931.36	−0.437458
$748$	0	0
$749$	−19361.5	−0.944530
$750$	605.846	0.0294965
$751$	−38229.4	−1.85754	−0.928769	−	0.370660i	$-0.879132\pi$
$751$	−38229.4	−1.85754	−0.928769	+	0.370660i	$0.879132\pi$
$752$	3375.30	0.163676
$753$	15114.1	0.731460
$754$	−1663.36	−0.0803395
$755$	−6027.54	−0.290549
$756$	−1691.61	−0.0813799
$757$	−33078.8	−1.58820	−0.794100	−	0.607787i	$-0.792058\pi$
$757$	−33078.8	−1.58820	−0.794100	+	0.607787i	$0.792058\pi$
$758$	8059.58	0.386197
$759$	0	0
$760$	2446.45	0.116766
$761$	−26978.2	−1.28510	−0.642548	−	0.766245i	$-0.722123\pi$
$761$	−26978.2	−1.28510	−0.642548	+	0.766245i	$0.722123\pi$
$762$	9892.67	0.470306
$763$	−26228.7	−1.24448
$764$	−1764.55	−0.0835593
$765$	3661.58	0.173052
$766$	18155.8	0.856391
$767$	−3062.67	−0.144181
$768$	11031.0	0.518289
$769$	14197.4	0.665763	0.332881	−	0.942969i	$-0.391979\pi$
$769$	14197.4	0.665763	0.332881	+	0.942969i	$0.391979\pi$
$770$	0	0
$771$	−8930.54	−0.417154
$772$	−21635.1	−1.00863
$773$	15313.6	0.712536	0.356268	−	0.934384i	$-0.384049\pi$
$773$	15313.6	0.712536	0.356268	+	0.934384i	$0.384049\pi$
$774$	−2647.55	−0.122951
$775$	−200.000	−0.00926995
$776$	−13037.5	−0.603116
$777$	3968.36	0.183223
$778$	−7897.89	−0.363950
$779$	−4716.40	−0.216922
$780$	−4437.68	−0.203711
$781$	0	0
$782$	19750.0	0.903144
$783$	506.445	0.0231148
$784$	−1698.31	−0.0773645
$785$	−17916.2	−0.814593
$786$	9998.64	0.453740
$787$	−15771.9	−0.714369	−0.357185	−	0.934034i	$-0.616263\pi$
$787$	−15771.9	−0.714369	−0.357185	+	0.934034i	$0.616263\pi$
$788$	20206.4	0.913481
$789$	−3658.07	−0.165058
$790$	−1234.29	−0.0555876
$791$	6763.67	0.304031
$792$	0	0
$793$	−12834.3	−0.574730
$794$	−15955.9	−0.713164
$795$	−3337.14	−0.148875
$796$	−18049.0	−0.803682
$797$	35431.8	1.57473	0.787365	−	0.616487i	$-0.211445\pi$
$797$	35431.8	1.57473	0.787365	+	0.616487i	$0.211445\pi$
$798$	−1274.30	−0.0565283
$799$	33617.7	1.48850
$800$	−4656.48	−0.205789
$801$	−3664.39	−0.161641
$802$	2672.17	0.117653
$803$	0	0
$804$	4434.32	0.194510
$805$	8731.88	0.382308
$806$	−709.427	−0.0310031
$807$	15107.8	0.659008
$808$	−18822.9	−0.819539
$809$	33730.8	1.46590	0.732948	−	0.680284i	$-0.238144\pi$
$809$	33730.8	1.46590	0.732948	+	0.680284i	$0.238144\pi$
$810$	−654.314	−0.0283830
$811$	−22454.5	−0.972237	−0.486119	−	0.873893i	$-0.661588\pi$
$811$	−22454.5	−0.972237	−0.486119	+	0.873893i	$0.661588\pi$
$812$	−1175.18	−0.0507891
$813$	3926.87	0.169399
$814$	0	0
$815$	−14100.7	−0.606044
$816$	−1994.24	−0.0855545
$817$	4118.40	0.176358
$818$	12193.3	0.521186
$819$	5742.32	0.244997
$820$	5619.52	0.239319
$821$	−29230.1	−1.24256	−0.621278	−	0.783590i	$-0.713386\pi$
$821$	−29230.1	−1.24256	−0.621278	+	0.783590i	$0.713386\pi$
$822$	−5435.13	−0.230623
$823$	10246.6	0.433990	0.216995	−	0.976173i	$-0.430374\pi$
$823$	10246.6	0.433990	0.216995	+	0.976173i	$0.430374\pi$
$824$	28115.1	1.18863
$825$	0	0
$826$	1047.86	0.0441401
$827$	−25079.5	−1.05453	−0.527267	−	0.849700i	$-0.676783\pi$
$827$	−25079.5	−1.05453	−0.527267	+	0.849700i	$0.676783\pi$
$828$	7287.86	0.305882
$829$	−37168.2	−1.55718	−0.778591	−	0.627531i	$-0.784065\pi$
$829$	−37168.2	−1.55718	−0.778591	+	0.627531i	$0.784065\pi$
$830$	8016.34	0.335243
$831$	18283.6	0.763238
$832$	−12929.8	−0.538773
$833$	−16915.0	−0.703564
$834$	1818.12	0.0754872
$835$	−6272.01	−0.259942
$836$	0	0
$837$	216.000	0.00892001
$838$	10671.5	0.439905
$839$	−30619.8	−1.25997	−0.629984	−	0.776608i	$-0.716939\pi$
$839$	−30619.8	−1.25997	−0.629984	+	0.776608i	$0.716939\pi$
$840$	3771.87	0.154931
$841$	−24037.2	−0.985574
$842$	20077.1	0.821737
$843$	15225.4	0.622052
$844$	19712.2	0.803934
$845$	4079.10	0.166065
$846$	−6007.39	−0.244135
$847$	0	0
$848$	1817.54	0.0736020
$849$	−17889.7	−0.723171
$850$	−3286.45	−0.132617
$851$	−17096.7	−0.688680
$852$	−642.368	−0.0258300
$853$	9001.86	0.361334	0.180667	−	0.983544i	$-0.442174\pi$
$853$	9001.86	0.361334	0.180667	+	0.983544i	$0.442174\pi$
$854$	4391.13	0.175950
$855$	1017.82	0.0407120
$856$	36032.0	1.43872
$857$	28946.8	1.15380	0.576899	−	0.816815i	$-0.304262\pi$
$857$	28946.8	1.15380	0.576899	+	0.816815i	$0.304262\pi$
$858$	0	0
$859$	36157.5	1.43618	0.718091	−	0.695950i	$-0.245016\pi$
$859$	36157.5	1.43618	0.718091	+	0.695950i	$0.245016\pi$
$860$	−4907.02	−0.194567
$861$	−7271.60	−0.287823
$862$	−5457.39	−0.215637
$863$	49979.4	1.97140	0.985701	−	0.168506i	$-0.0538944\pi$
$863$	49979.4	1.97140	0.985701	+	0.168506i	$0.0538944\pi$
$864$	5029.00	0.198021
$865$	11235.1	0.441626
$866$	11265.3	0.442045
$867$	−5123.48	−0.200695
$868$	−501.217	−0.0195996
$869$	0	0
$870$	−454.560	−0.0177138
$871$	−15052.7	−0.585581
$872$	48811.9	1.89562
$873$	−5424.11	−0.210284
$874$	5489.97	0.212473
$875$	−1453.01	−0.0561379
$876$	10077.7	0.388692
$877$	49073.8	1.88952	0.944758	−	0.327768i	$-0.106296\pi$
$877$	49073.8	1.88952	0.944758	+	0.327768i	$0.106296\pi$
$878$	8806.03	0.338484
$879$	−4267.83	−0.163766
$880$	0	0
$881$	−47103.5	−1.80131	−0.900657	−	0.434531i	$-0.856914\pi$
$881$	−47103.5	−1.80131	−0.900657	+	0.434531i	$0.856914\pi$
$882$	3022.66	0.115395
$883$	−17697.8	−0.674494	−0.337247	−	0.941416i	$-0.609496\pi$
$883$	−17697.8	−0.674494	−0.337247	+	0.941416i	$0.609496\pi$
$884$	24072.5	0.915888
$885$	−836.960	−0.0317899
$886$	2328.80	0.0883042
$887$	−28545.1	−1.08055	−0.540277	−	0.841487i	$-0.681680\pi$
$887$	−28545.1	−1.08055	−0.540277	+	0.841487i	$0.681680\pi$
$888$	−7385.18	−0.279088
$889$	−23725.7	−0.895089
$890$	3288.97	0.123873
$891$	0	0
$892$	25424.2	0.954334
$893$	9344.83	0.350182
$894$	2501.48	0.0935818
$895$	7395.99	0.276224
$896$	−12896.9	−0.480866
$897$	−24739.3	−0.920870
$898$	9790.31	0.363816
$899$	150.058	0.00556697
$900$	−1212.72	−0.0449156
$901$	18102.5	0.669347
$902$	0	0
$903$	6349.64	0.234001
$904$	−12587.3	−0.463105
$905$	−13611.7	−0.499966
$906$	−5842.82	−0.214255
$907$	37690.7	1.37982	0.689911	−	0.723894i	$-0.257650\pi$
$907$	37690.7	1.37982	0.689911	+	0.723894i	$0.257650\pi$
$908$	30487.4	1.11427
$909$	−7831.08	−0.285743
$910$	−5154.02	−0.187752
$911$	43096.8	1.56736	0.783678	−	0.621168i	$-0.213341\pi$
$911$	43096.8	1.56736	0.783678	+	0.621168i	$0.213341\pi$
$912$	−554.347	−0.0201275
$913$	0	0
$914$	−14142.3	−0.511799
$915$	−3507.34	−0.126720
$916$	31675.4	1.14256
$917$	−23979.9	−0.863560
$918$	3549.37	0.127611
$919$	11040.0	0.396275	0.198137	−	0.980174i	$-0.436511\pi$
$919$	11040.0	0.396275	0.198137	+	0.980174i	$0.436511\pi$
$920$	−16250.1	−0.582338
$921$	−23390.6	−0.836857
$922$	−21821.8	−0.779462
$923$	2180.58	0.0777623
$924$	0	0
$925$	2844.93	0.101125
$926$	−12457.5	−0.442093
$927$	11697.0	0.414433
$928$	3493.70	0.123585
$929$	43768.7	1.54575	0.772877	−	0.634556i	$-0.218817\pi$
$929$	43768.7	1.54575	0.772877	+	0.634556i	$0.218817\pi$
$930$	−193.871	−0.00683578
$931$	−4701.91	−0.165520
$932$	−22634.5	−0.795514
$933$	9643.56	0.338388
$934$	−31618.7	−1.10770
$935$	0	0
$936$	−10686.5	−0.373184
$937$	−4092.06	−0.142670	−0.0713351	−	0.997452i	$-0.522726\pi$
$937$	−4092.06	−0.142670	−0.0713351	+	0.997452i	$0.522726\pi$
$938$	5150.11	0.179272
$939$	407.632	0.0141667
$940$	−11134.2	−0.386339
$941$	21916.6	0.759255	0.379628	−	0.925139i	$-0.376052\pi$
$941$	21916.6	0.759255	0.379628	+	0.925139i	$0.376052\pi$
$942$	−17367.1	−0.600691
$943$	31327.8	1.08184
$944$	455.842	0.0157165
$945$	1569.25	0.0540187
$946$	0	0
$947$	5474.59	0.187857	0.0939284	−	0.995579i	$-0.470058\pi$
$947$	5474.59	0.187857	0.0939284	+	0.995579i	$0.470058\pi$
$948$	2470.68	0.0846455
$949$	−34209.6	−1.17017
$950$	−913.547	−0.0311993
$951$	20570.4	0.701408
$952$	−20460.8	−0.696573
$953$	18889.2	0.642057	0.321028	−	0.947070i	$-0.395972\pi$
$953$	18889.2	0.642057	0.321028	+	0.947070i	$0.395972\pi$
$954$	−3234.87	−0.109783
$955$	1636.92	0.0554653
$956$	−20434.3	−0.691311
$957$	0	0
$958$	27670.3	0.933182
$959$	13035.1	0.438922
$960$	−3533.42	−0.118792
$961$	−29727.0	−0.997852
$962$	10091.4	0.338211
$963$	14990.7	0.501630
$964$	29515.3	0.986124
$965$	20070.1	0.669513
$966$	8464.28	0.281919
$967$	1527.63	0.0508018	0.0254009	−	0.999677i	$-0.491914\pi$
$967$	1527.63	0.0508018	0.0254009	+	0.999677i	$0.491914\pi$
$968$	0	0
$969$	−5521.24	−0.183042
$970$	4868.41	0.161150
$971$	−18322.1	−0.605545	−0.302772	−	0.953063i	$-0.597912\pi$
$971$	−18322.1	−0.605545	−0.302772	+	0.953063i	$0.597912\pi$
$972$	1309.74	0.0432200
$973$	−4360.41	−0.143667
$974$	5499.56	0.180921
$975$	4116.69	0.135220
$976$	1910.24	0.0626489
$977$	15644.7	0.512302	0.256151	−	0.966637i	$-0.417546\pi$
$977$	15644.7	0.512302	0.256151	+	0.966637i	$0.417546\pi$
$978$	−13668.6	−0.446905
$979$	0	0
$980$	5602.26	0.182610
$981$	20307.7	0.660933
$982$	−7666.22	−0.249123
$983$	7449.41	0.241708	0.120854	−	0.992670i	$-0.461437\pi$
$983$	7449.41	0.241708	0.120854	+	0.992670i	$0.461437\pi$
$984$	13532.6	0.438417
$985$	−18744.8	−0.606355
$986$	2465.79	0.0796417
$987$	14407.6	0.464639
$988$	6691.51	0.215471
$989$	−27355.8	−0.879538
$990$	0	0
$991$	−4489.23	−0.143900	−0.0719501	−	0.997408i	$-0.522922\pi$
$991$	−4489.23	−0.143900	−0.0719501	+	0.997408i	$0.522922\pi$
$992$	1490.07	0.0476914
$993$	17930.3	0.573013
$994$	−746.061	−0.0238065
$995$	16743.5	0.533471
$996$	−16046.3	−0.510488
$997$	−37438.8	−1.18927	−0.594634	−	0.803997i	$-0.702703\pi$
$997$	−37438.8	−1.18927	−0.594634	+	0.803997i	$0.702703\pi$
$998$	18603.1	0.590052
$999$	−3072.53	−0.0973078

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.v.1.2	✓	4
11.10	odd	2	inner	1815.4.a.v.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.4.a.v.1.2	✓	4	1.1	even	1	trivial
1815.4.a.v.1.3	yes	4	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)

\(f(q)\)	\(=\)	\(q-1.61559 q^{2} -3.00000 q^{3} -5.38987 q^{4} +5.00000 q^{5} +4.84677 q^{6} -11.6241 q^{7} +21.6325 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.61559 q^{2} -3.00000 q^{3} -5.38987 q^{4} +5.00000 q^{5} +4.84677 q^{6} -11.6241 q^{7} +21.6325 q^{8} +9.00000 q^{9} -8.07795 q^{10} +16.1696 q^{12} -54.8892 q^{13} +18.7797 q^{14} -15.0000 q^{15} +8.16960 q^{16} +81.3685 q^{17} -14.5403 q^{18} +22.6183 q^{19} -26.9493 q^{20} +34.8722 q^{21} -150.238 q^{23} -64.8976 q^{24} +25.0000 q^{25} +88.6784 q^{26} -27.0000 q^{27} +62.6522 q^{28} -18.7572 q^{29} +24.2339 q^{30} -8.00000 q^{31} -186.259 q^{32} -131.458 q^{34} -58.1203 q^{35} -48.5088 q^{36} +113.797 q^{37} -36.5419 q^{38} +164.667 q^{39} +108.163 q^{40} -208.521 q^{41} -56.3392 q^{42} +182.083 q^{43} +45.0000 q^{45} +242.723 q^{46} +413.154 q^{47} -24.5088 q^{48} -207.881 q^{49} -40.3898 q^{50} -244.105 q^{51} +295.845 q^{52} +222.476 q^{53} +43.6209 q^{54} -251.458 q^{56} -67.8548 q^{57} +30.3040 q^{58} +55.7973 q^{59} +80.8480 q^{60} +233.823 q^{61} +12.9247 q^{62} -104.617 q^{63} +235.562 q^{64} -274.446 q^{65} +274.238 q^{67} -438.565 q^{68} +450.714 q^{69} +93.8987 q^{70} -39.7269 q^{71} +194.693 q^{72} +623.250 q^{73} -183.850 q^{74} -75.0000 q^{75} -121.909 q^{76} -266.035 q^{78} +152.798 q^{79} +40.8480 q^{80} +81.0000 q^{81} +336.885 q^{82} -992.373 q^{83} -187.957 q^{84} +406.842 q^{85} -294.172 q^{86} +56.2717 q^{87} -407.154 q^{89} -72.7016 q^{90} +638.035 q^{91} +809.762 q^{92} +24.0000 q^{93} -667.488 q^{94} +113.091 q^{95} +558.777 q^{96} -602.678 q^{97} +335.851 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 10 q^{4} + 20 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 10 * q^4 + 20 * q^5 + 36 * q^9	\( 4 q - 12 q^{3} + 10 q^{4} + 20 q^{5} + 36 q^{9} - 30 q^{12} + 12 q^{14} - 60 q^{15} - 62 q^{16} + 50 q^{20} - 96 q^{23} + 100 q^{25} - 24 q^{26} - 108 q^{27} - 32 q^{31} - 84 q^{34} + 90 q^{36} - 176 q^{37} - 588 q^{38} - 36 q^{42} + 180 q^{45} + 264 q^{47} + 186 q^{48} - 1084 q^{49} - 120 q^{53} - 564 q^{56} + 1068 q^{58} - 408 q^{59} - 150 q^{60} - 1046 q^{64} + 592 q^{67} + 288 q^{69} + 60 q^{70} - 1800 q^{71} - 300 q^{75} + 72 q^{78} - 310 q^{80} + 324 q^{81} - 2124 q^{82} + 780 q^{86} - 240 q^{89} + 1416 q^{91} + 3744 q^{92} + 96 q^{93} - 2032 q^{97}+O(q^{100}) \) 4 * q - 12 * q^3 + 10 * q^4 + 20 * q^5 + 36 * q^9 - 30 * q^12 + 12 * q^14 - 60 * q^15 - 62 * q^16 + 50 * q^20 - 96 * q^23 + 100 * q^25 - 24 * q^26 - 108 * q^27 - 32 * q^31 - 84 * q^34 + 90 * q^36 - 176 * q^37 - 588 * q^38 - 36 * q^42 + 180 * q^45 + 264 * q^47 + 186 * q^48 - 1084 * q^49 - 120 * q^53 - 564 * q^56 + 1068 * q^58 - 408 * q^59 - 150 * q^60 - 1046 * q^64 + 592 * q^67 + 288 * q^69 + 60 * q^70 - 1800 * q^71 - 300 * q^75 + 72 * q^78 - 310 * q^80 + 324 * q^81 - 2124 * q^82 + 780 * q^86 - 240 * q^89 + 1416 * q^91 + 3744 * q^92 + 96 * q^93 - 2032 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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