LMFDB - Embedded newform 605.4.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,4,Mod(1,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("605.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	605.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:	$35.6961555535$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 67 x^{10} + 65 x^{9} + 1558 x^{8} - 1475 x^{7} - 14915 x^{6} + 12951 x^{5} + \cdots + 27856 $ x^12 - x^11 - 67x^10 + 65x^9 + 1558x^8 - 1475x^7 - 14915x^6 + 12951x^5 + 54155x^4 - 29108x^3 - 78796x^2 + 10176x + 27856
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.36383$ of defining polynomial
Character	$\chi$	$=$	605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.36383 q^{2} -6.47441 q^{3} +11.0430 q^{4} -5.00000 q^{5} +28.2532 q^{6} +24.2107 q^{7} -13.2791 q^{8} +14.9180 q^{9} +21.8191 q^{10} -71.4969 q^{12} -71.3811 q^{13} -105.652 q^{14} +32.3720 q^{15} -30.3961 q^{16} +51.8749 q^{17} -65.0994 q^{18} -87.9570 q^{19} -55.2150 q^{20} -156.750 q^{21} +43.9127 q^{23} +85.9745 q^{24} +25.0000 q^{25} +311.495 q^{26} +78.2241 q^{27} +267.359 q^{28} -166.587 q^{29} -141.266 q^{30} +224.856 q^{31} +238.877 q^{32} -226.373 q^{34} -121.054 q^{35} +164.739 q^{36} +19.7264 q^{37} +383.829 q^{38} +462.151 q^{39} +66.3957 q^{40} +131.128 q^{41} +684.031 q^{42} +109.681 q^{43} -74.5898 q^{45} -191.627 q^{46} +268.086 q^{47} +196.797 q^{48} +243.160 q^{49} -109.096 q^{50} -335.860 q^{51} -788.262 q^{52} -82.4950 q^{53} -341.356 q^{54} -321.498 q^{56} +569.469 q^{57} +726.956 q^{58} -622.051 q^{59} +357.484 q^{60} -261.622 q^{61} -981.234 q^{62} +361.175 q^{63} -799.248 q^{64} +356.906 q^{65} +939.377 q^{67} +572.855 q^{68} -284.309 q^{69} +528.258 q^{70} +1084.62 q^{71} -198.098 q^{72} -339.471 q^{73} -86.0826 q^{74} -161.860 q^{75} -971.309 q^{76} -2016.75 q^{78} -1325.47 q^{79} +151.981 q^{80} -909.239 q^{81} -572.219 q^{82} +1363.89 q^{83} -1730.99 q^{84} -259.375 q^{85} -478.630 q^{86} +1078.55 q^{87} +664.950 q^{89} +325.497 q^{90} -1728.19 q^{91} +484.928 q^{92} -1455.81 q^{93} -1169.88 q^{94} +439.785 q^{95} -1546.58 q^{96} -775.214 q^{97} -1061.11 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + q^{3} + 39 q^{4} - 60 q^{5} - 10 q^{6} - 41 q^{7} + 9 q^{8} + 131 q^{9} + 5 q^{10} - 29 q^{12} - 109 q^{13} - 247 q^{14} - 5 q^{15} + 283 q^{16} + 167 q^{17} - 135 q^{18} - 332 q^{19} - 195 q^{20}+ \cdots + 4209 q^{98}+O(q^{100}) $ 12 * q - q^2 + q^3 + 39 * q^4 - 60 * q^5 - 10 * q^6 - 41 * q^7 + 9 * q^8 + 131 * q^9 + 5 * q^10 - 29 * q^12 - 109 * q^13 - 247 * q^14 - 5 * q^15 + 283 * q^16 + 167 * q^17 - 135 * q^18 - 332 * q^19 - 195 * q^20 - 204 * q^21 - 193 * q^23 - 444 * q^24 + 300 * q^25 + 546 * q^26 + 334 * q^27 - 244 * q^28 - 216 * q^29 + 50 * q^30 - 252 * q^31 - 234 * q^32 - 665 * q^34 + 205 * q^35 - 473 * q^36 - 45 * q^37 - 238 * q^38 - 1144 * q^39 - 45 * q^40 - 998 * q^41 - 248 * q^42 - 371 * q^43 - 655 * q^45 - 2567 * q^46 + 257 * q^47 + 1605 * q^48 + 1057 * q^49 - 25 * q^50 - 2064 * q^51 - 3488 * q^52 - 201 * q^53 - 1140 * q^54 - 2710 * q^56 - 294 * q^57 + 1090 * q^58 + 20 * q^59 + 145 * q^60 - 1662 * q^61 - 737 * q^62 - 1911 * q^63 + 527 * q^64 + 545 * q^65 - 1941 * q^67 + 1468 * q^68 + 2178 * q^69 + 1235 * q^70 + 1032 * q^71 - 3514 * q^72 - 2441 * q^73 - 2697 * q^74 + 25 * q^75 - 3852 * q^76 - 3166 * q^78 - 2406 * q^79 - 1415 * q^80 - 204 * q^81 + 2956 * q^82 + 953 * q^83 - 4269 * q^84 - 835 * q^85 - 692 * q^86 - 910 * q^87 - 4204 * q^89 + 675 * q^90 - 328 * q^91 + 6565 * q^92 + 1630 * q^93 - 1196 * q^94 + 1660 * q^95 - 6253 * q^96 + 543 * q^97 + 4209 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.36383	−1.54285	−0.771423	−	0.636322i	$-0.780455\pi$
$2$	−4.36383	−1.54285	−0.771423	+	0.636322i	$0.780455\pi$
$3$	−6.47441	−1.24600	−0.623000	−	0.782222i	$-0.714086\pi$
$3$	−6.47441	−1.24600	−0.623000	+	0.782222i	$0.714086\pi$
$4$	11.0430	1.38038
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	28.2532	1.92239
$7$	24.2107	1.30726	0.653628	−	0.756816i	$-0.273246\pi$
$7$	24.2107	1.30726	0.653628	+	0.756816i	$0.273246\pi$
$8$	−13.2791	−0.586860
$9$	14.9180	0.552517
$10$	21.8191	0.689982
$11$	0	0
$11$	0	0
$12$	−71.4969	−1.71995
$13$	−71.3811	−1.52289	−0.761445	−	0.648230i	$-0.775510\pi$
$13$	−71.3811	−1.52289	−0.761445	+	0.648230i	$0.775510\pi$
$14$	−105.652	−2.01690
$15$	32.3720	0.557228
$16$	−30.3961	−0.474940
$17$	51.8749	0.740090	0.370045	−	0.929014i	$-0.379342\pi$
$17$	51.8749	0.740090	0.370045	+	0.929014i	$0.379342\pi$
$18$	−65.0994	−0.852449
$19$	−87.9570	−1.06204	−0.531019	−	0.847360i	$-0.678191\pi$
$19$	−87.9570	−1.06204	−0.531019	+	0.847360i	$0.678191\pi$
$20$	−55.2150	−0.617323
$21$	−156.750	−1.62884
$22$	0	0
$23$	43.9127	0.398106	0.199053	−	0.979989i	$-0.436213\pi$
$23$	43.9127	0.398106	0.199053	+	0.979989i	$0.436213\pi$
$24$	85.9745	0.731228
$25$	25.0000	0.200000
$26$	311.495	2.34959
$27$	78.2241	0.557564
$28$	267.359	1.80450
$29$	−166.587	−1.06670	−0.533351	−	0.845894i	$-0.679068\pi$
$29$	−166.587	−1.06670	−0.533351	+	0.845894i	$0.679068\pi$
$30$	−141.266	−0.859718
$31$	224.856	1.30275	0.651377	−	0.758754i	$-0.274192\pi$
$31$	224.856	1.30275	0.651377	+	0.758754i	$0.274192\pi$
$32$	238.877	1.31962
$33$	0	0
$34$	−226.373	−1.14184
$35$	−121.054	−0.584623
$36$	164.739	0.762681
$37$	19.7264	0.0876486	0.0438243	−	0.999039i	$-0.486046\pi$
$37$	19.7264	0.0876486	0.0438243	+	0.999039i	$0.486046\pi$
$38$	383.829	1.63856
$39$	462.151	1.89752
$40$	66.3957	0.262452
$41$	131.128	0.499481	0.249741	−	0.968313i	$-0.419655\pi$
$41$	131.128	0.499481	0.249741	+	0.968313i	$0.419655\pi$
$42$	684.031	2.51305
$43$	109.681	0.388982	0.194491	−	0.980904i	$-0.437694\pi$
$43$	109.681	0.388982	0.194491	+	0.980904i	$0.437694\pi$
$44$	0	0
$45$	−74.5898	−0.247093
$46$	−191.627	−0.614216
$47$	268.086	0.832009	0.416005	−	0.909363i	$-0.363430\pi$
$47$	268.086	0.832009	0.416005	+	0.909363i	$0.363430\pi$
$48$	196.797	0.591775
$49$	243.160	0.708920
$50$	−109.096	−0.308569
$51$	−335.860	−0.922152
$52$	−788.262	−2.10216
$53$	−82.4950	−0.213803	−0.106902	−	0.994270i	$-0.534093\pi$
$53$	−82.4950	−0.213803	−0.106902	+	0.994270i	$0.534093\pi$
$54$	−341.356	−0.860236
$55$	0	0
$56$	−321.498	−0.767177
$57$	569.469	1.32330
$58$	726.956	1.64576
$59$	−622.051	−1.37261	−0.686306	−	0.727313i	$-0.740769\pi$
$59$	−622.051	−1.37261	−0.686306	+	0.727313i	$0.740769\pi$
$60$	357.484	0.769184
$61$	−261.622	−0.549136	−0.274568	−	0.961568i	$-0.588535\pi$
$61$	−261.622	−0.549136	−0.274568	+	0.961568i	$0.588535\pi$
$62$	−981.234	−2.00995
$63$	361.175	0.722282
$64$	−799.248	−1.56103
$65$	356.906	0.681057
$66$	0	0
$67$	939.377	1.71288	0.856441	−	0.516244i	$-0.172670\pi$
$67$	939.377	1.71288	0.856441	+	0.516244i	$0.172670\pi$
$68$	572.855	1.02160
$69$	−284.309	−0.496040
$70$	528.258	0.901984
$71$	1084.62	1.81297	0.906484	−	0.422241i	$-0.138756\pi$
$71$	1084.62	1.81297	0.906484	+	0.422241i	$0.138756\pi$
$72$	−198.098	−0.324250
$73$	−339.471	−0.544276	−0.272138	−	0.962258i	$-0.587731\pi$
$73$	−339.471	−0.544276	−0.272138	+	0.962258i	$0.587731\pi$
$74$	−86.0826	−0.135228
$75$	−161.860	−0.249200
$76$	−971.309	−1.46601
$77$	0	0
$78$	−2016.75	−2.92758
$79$	−1325.47	−1.88769	−0.943843	−	0.330395i	$-0.892818\pi$
$79$	−1325.47	−1.88769	−0.943843	+	0.330395i	$0.892818\pi$
$80$	151.981	0.212399
$81$	−909.239	−1.24724
$82$	−572.219	−0.770623
$83$	1363.89	1.80370	0.901848	−	0.432053i	$-0.142211\pi$
$83$	1363.89	1.80370	0.901848	+	0.432053i	$0.142211\pi$
$84$	−1730.99	−2.24841
$85$	−259.375	−0.330978
$86$	−478.630	−0.600140
$87$	1078.55	1.32911
$88$	0	0
$89$	664.950	0.791961	0.395981	−	0.918259i	$-0.370405\pi$
$89$	664.950	0.791961	0.395981	+	0.918259i	$0.370405\pi$
$90$	325.497	0.381227
$91$	−1728.19	−1.99081
$92$	484.928	0.549535
$93$	−1455.81	−1.62323
$94$	−1169.88	−1.28366
$95$	439.785	0.474958
$96$	−1546.58	−1.64425
$97$	−775.214	−0.811455	−0.405727	−	0.913994i	$-0.632982\pi$
$97$	−775.214	−0.811455	−0.405727	+	0.913994i	$0.632982\pi$
$98$	−1061.11	−1.09376
$99$	0	0
$100$	276.075	0.276075
$101$	1572.80	1.54950	0.774748	−	0.632270i	$-0.217877\pi$
$101$	1572.80	1.54950	0.774748	+	0.632270i	$0.217877\pi$
$102$	1465.63	1.42274
$103$	−34.7443	−0.0332375	−0.0166187	−	0.999862i	$-0.505290\pi$
$103$	−34.7443	−0.0332375	−0.0166187	+	0.999862i	$0.505290\pi$
$104$	947.880	0.893724
$105$	783.751	0.728441
$106$	359.994	0.329865
$107$	369.753	0.334069	0.167034	−	0.985951i	$-0.446581\pi$
$107$	369.753	0.334069	0.167034	+	0.985951i	$0.446581\pi$
$108$	863.829	0.769648
$109$	−543.635	−0.477713	−0.238857	−	0.971055i	$-0.576773\pi$
$109$	−543.635	−0.477713	−0.238857	+	0.971055i	$0.576773\pi$
$110$	0	0
$111$	−127.717	−0.109210
$112$	−735.913	−0.620868
$113$	−230.058	−0.191522	−0.0957611	−	0.995404i	$-0.530528\pi$
$113$	−230.058	−0.191522	−0.0957611	+	0.995404i	$0.530528\pi$
$114$	−2485.07	−2.04165
$115$	−219.563	−0.178038
$116$	−1839.62	−1.47245
$117$	−1064.86	−0.841422
$118$	2714.52	2.11773
$119$	1255.93	0.967487
$120$	−429.873	−0.327015
$121$	0	0
$122$	1141.68	0.847233
$123$	−848.975	−0.622354
$124$	2483.09	1.79829
$125$	−125.000	−0.0894427
$126$	−1576.10	−1.11437
$127$	−2050.29	−1.43255	−0.716277	−	0.697817i	$-0.754155\pi$
$127$	−2050.29	−1.43255	−0.716277	+	0.697817i	$0.754155\pi$
$128$	1576.77	1.08881
$129$	−710.121	−0.484672
$130$	−1557.48	−1.05077
$131$	−2484.58	−1.65709	−0.828545	−	0.559922i	$-0.810831\pi$
$131$	−2484.58	−1.65709	−0.828545	+	0.559922i	$0.810831\pi$
$132$	0	0
$133$	−2129.50	−1.38836
$134$	−4099.28	−2.64272
$135$	−391.120	−0.249350
$136$	−688.854	−0.434329
$137$	−54.1791	−0.0337871	−0.0168935	−	0.999857i	$-0.505378\pi$
$137$	−54.1791	−0.0337871	−0.0168935	+	0.999857i	$0.505378\pi$
$138$	1240.67	0.765313
$139$	228.690	0.139549	0.0697744	−	0.997563i	$-0.477772\pi$
$139$	228.690	0.139549	0.0697744	+	0.997563i	$0.477772\pi$
$140$	−1336.80	−0.806999
$141$	−1735.70	−1.03668
$142$	−4733.09	−2.79713
$143$	0	0
$144$	−453.448	−0.262412
$145$	832.933	0.477044
$146$	1481.40	0.839734
$147$	−1574.32	−0.883315
$148$	217.839	0.120988
$149$	−1221.62	−0.671673	−0.335836	−	0.941920i	$-0.609019\pi$
$149$	−1221.62	−0.671673	−0.335836	+	0.941920i	$0.609019\pi$
$150$	706.330	0.384477
$151$	−440.911	−0.237621	−0.118811	−	0.992917i	$-0.537908\pi$
$151$	−440.911	−0.237621	−0.118811	+	0.992917i	$0.537908\pi$
$152$	1167.99	0.623268
$153$	773.868	0.408912
$154$	0	0
$155$	−1124.28	−0.582609
$156$	5103.53	2.61929
$157$	−22.3192	−0.0113456	−0.00567282	−	0.999984i	$-0.501806\pi$
$157$	−22.3192	−0.0113456	−0.00567282	+	0.999984i	$0.501806\pi$
$158$	5784.13	2.91241
$159$	534.107	0.266399
$160$	−1194.38	−0.590152
$161$	1063.16	0.520426
$162$	3967.76	1.92430
$163$	−2648.57	−1.27271	−0.636355	−	0.771396i	$-0.719559\pi$
$163$	−2648.57	−1.27271	−0.636355	+	0.771396i	$0.719559\pi$
$164$	1448.04	0.689471
$165$	0	0
$166$	−5951.80	−2.78283
$167$	−19.4773	−0.00902516	−0.00451258	−	0.999990i	$-0.501436\pi$
$167$	−19.4773	−0.00902516	−0.00451258	+	0.999990i	$0.501436\pi$
$168$	2081.51	0.955903
$169$	2898.27	1.31919
$170$	1131.87	0.510648
$171$	−1312.14	−0.586794
$172$	1211.21	0.536941
$173$	−2667.14	−1.17213	−0.586066	−	0.810263i	$-0.699324\pi$
$173$	−2667.14	−1.17213	−0.586066	+	0.810263i	$0.699324\pi$
$174$	−4706.61	−2.05062
$175$	605.268	0.261451
$176$	0	0
$177$	4027.41	1.71027
$178$	−2901.73	−1.22187
$179$	−91.6342	−0.0382629	−0.0191315	−	0.999817i	$-0.506090\pi$
$179$	−91.6342	−0.0382629	−0.0191315	+	0.999817i	$0.506090\pi$
$180$	−823.695	−0.341081
$181$	−2750.79	−1.12964	−0.564819	−	0.825215i	$-0.691054\pi$
$181$	−2750.79	−1.12964	−0.564819	+	0.825215i	$0.691054\pi$
$182$	7541.53	3.07151
$183$	1693.85	0.684224
$184$	−583.122	−0.233632
$185$	−98.6319	−0.0391976
$186$	6352.91	2.50440
$187$	0	0
$188$	2960.48	1.14848
$189$	1893.86	0.728879
$190$	−1919.15	−0.732787
$191$	2441.77	0.925028	0.462514	−	0.886612i	$-0.346947\pi$
$191$	2441.77	0.925028	0.462514	+	0.886612i	$0.346947\pi$
$192$	5174.65	1.94504
$193$	1436.59	0.535792	0.267896	−	0.963448i	$-0.413672\pi$
$193$	1436.59	0.535792	0.267896	+	0.963448i	$0.413672\pi$
$194$	3382.90	1.25195
$195$	−2310.75	−0.848597
$196$	2685.21	0.978576
$197$	2227.72	0.805677	0.402839	−	0.915271i	$-0.368024\pi$
$197$	2227.72	0.805677	0.402839	+	0.915271i	$0.368024\pi$
$198$	0	0
$199$	4105.01	1.46229	0.731146	−	0.682221i	$-0.238986\pi$
$199$	4105.01	1.46229	0.731146	+	0.682221i	$0.238986\pi$
$200$	−331.978	−0.117372
$201$	−6081.91	−2.13425
$202$	−6863.42	−2.39064
$203$	−4033.19	−1.39445
$204$	−3708.90	−1.27292
$205$	−655.639	−0.223375
$206$	151.618	0.0512803
$207$	655.088	0.219960
$208$	2169.71	0.723281
$209$	0	0
$210$	−3420.15	−1.12387
$211$	−4079.12	−1.33089	−0.665446	−	0.746446i	$-0.731759\pi$
$211$	−4079.12	−1.33089	−0.665446	+	0.746446i	$0.731759\pi$
$212$	−910.993	−0.295128
$213$	−7022.27	−2.25896
$214$	−1613.54	−0.515417
$215$	−548.406	−0.173958
$216$	−1038.75	−0.327212
$217$	5443.94	1.70303
$218$	2372.33	0.737038
$219$	2197.88	0.678168
$220$	0	0
$221$	−3702.89	−1.12707
$222$	557.334	0.168495
$223$	−177.793	−0.0533898	−0.0266949	−	0.999644i	$-0.508498\pi$
$223$	−177.793	−0.0533898	−0.0266949	+	0.999644i	$0.508498\pi$
$224$	5783.38	1.72508
$225$	372.949	0.110503
$226$	1003.93	0.295489
$227$	988.041	0.288892	0.144446	−	0.989513i	$-0.453860\pi$
$227$	988.041	0.288892	0.144446	+	0.989513i	$0.453860\pi$
$228$	6288.65	1.82665
$229$	−521.699	−0.150545	−0.0752726	−	0.997163i	$-0.523983\pi$
$229$	−521.699	−0.150545	−0.0752726	+	0.997163i	$0.523983\pi$
$230$	958.137	0.274686
$231$	0	0
$232$	2212.13	0.626005
$233$	3468.16	0.975136	0.487568	−	0.873085i	$-0.337884\pi$
$233$	3468.16	0.975136	0.487568	+	0.873085i	$0.337884\pi$
$234$	4646.87	1.29819
$235$	−1340.43	−0.372086
$236$	−6869.30	−1.89472
$237$	8581.64	2.35206
$238$	−5480.67	−1.49268
$239$	−4008.96	−1.08501	−0.542507	−	0.840051i	$-0.682525\pi$
$239$	−4008.96	−1.08501	−0.542507	+	0.840051i	$0.682525\pi$
$240$	−983.985	−0.264650
$241$	−1695.51	−0.453184	−0.226592	−	0.973990i	$-0.572758\pi$
$241$	−1695.51	−0.453184	−0.226592	+	0.973990i	$0.572758\pi$
$242$	0	0
$243$	3774.74	0.996500
$244$	−2889.10	−0.758014
$245$	−1215.80	−0.317039
$246$	3704.78	0.960196
$247$	6278.47	1.61737
$248$	−2985.90	−0.764535
$249$	−8830.40	−2.24741
$250$	545.479	0.137996
$251$	−3752.81	−0.943727	−0.471864	−	0.881672i	$-0.656419\pi$
$251$	−3752.81	−0.943727	−0.471864	+	0.881672i	$0.656419\pi$
$252$	3988.45	0.997019
$253$	0	0
$254$	8947.14	2.21021
$255$	1679.30	0.412399
$256$	−486.758	−0.118837
$257$	−2086.02	−0.506312	−0.253156	−	0.967425i	$-0.581469\pi$
$257$	−2086.02	−0.506312	−0.253156	+	0.967425i	$0.581469\pi$
$258$	3098.85	0.747774
$259$	477.590	0.114579
$260$	3941.31	0.940114
$261$	−2485.13	−0.589371
$262$	10842.3	2.55664
$263$	28.3895	0.00665617	0.00332809	−	0.999994i	$-0.498941\pi$
$263$	28.3895	0.00665617	0.00332809	+	0.999994i	$0.498941\pi$
$264$	0	0
$265$	412.475	0.0956157
$266$	9292.78	2.14202
$267$	−4305.16	−0.986784
$268$	10373.5	2.36442
$269$	1108.10	0.251159	0.125580	−	0.992084i	$-0.459921\pi$
$269$	1108.10	0.251159	0.125580	+	0.992084i	$0.459921\pi$
$270$	1706.78	0.384709
$271$	3007.39	0.674119	0.337059	−	0.941483i	$-0.390568\pi$
$271$	3007.39	0.674119	0.337059	+	0.941483i	$0.390568\pi$
$272$	−1576.80	−0.351498
$273$	11189.0	2.48055
$274$	236.428	0.0521283
$275$	0	0
$276$	−3139.62	−0.684721
$277$	−7694.70	−1.66906	−0.834531	−	0.550962i	$-0.814261\pi$
$277$	−7694.70	−1.66906	−0.834531	+	0.550962i	$0.814261\pi$
$278$	−997.966	−0.215302
$279$	3354.40	0.719794
$280$	1607.49	0.343092
$281$	−4032.39	−0.856057	−0.428029	−	0.903765i	$-0.640792\pi$
$281$	−4032.39	−0.856057	−0.428029	+	0.903765i	$0.640792\pi$
$282$	7574.30	1.59944
$283$	1462.66	0.307231	0.153615	−	0.988131i	$-0.450908\pi$
$283$	1462.66	0.307231	0.153615	+	0.988131i	$0.450908\pi$
$284$	11977.5	2.50258
$285$	−2847.35	−0.591797
$286$	0	0
$287$	3174.70	0.652950
$288$	3563.55	0.729112
$289$	−2221.99	−0.452267
$290$	−3634.78	−0.736005
$291$	5019.05	1.01107
$292$	−3748.78	−0.751305
$293$	−5268.27	−1.05043	−0.525215	−	0.850970i	$-0.676015\pi$
$293$	−5268.27	−1.05043	−0.525215	+	0.850970i	$0.676015\pi$
$294$	6870.04	1.36282
$295$	3110.25	0.613851
$296$	−261.949	−0.0514375
$297$	0	0
$298$	5330.95	1.03629
$299$	−3134.54	−0.606271
$300$	−1787.42	−0.343990
$301$	2655.46	0.508499
$302$	1924.06	0.366613
$303$	−10182.9	−1.93067
$304$	2673.55	0.504404
$305$	1308.11	0.245581
$306$	−3377.03	−0.630888
$307$	−5709.75	−1.06148	−0.530738	−	0.847536i	$-0.678085\pi$
$307$	−5709.75	−1.06148	−0.530738	+	0.847536i	$0.678085\pi$
$308$	0	0
$309$	224.949	0.0414139
$310$	4906.17	0.898877
$311$	−3313.49	−0.604151	−0.302075	−	0.953284i	$-0.597679\pi$
$311$	−3313.49	−0.604151	−0.302075	+	0.953284i	$0.597679\pi$
$312$	−6136.96	−1.11358
$313$	−6393.18	−1.15452	−0.577259	−	0.816561i	$-0.695878\pi$
$313$	−6393.18	−1.15452	−0.577259	+	0.816561i	$0.695878\pi$
$314$	97.3971	0.0175046
$315$	−1805.87	−0.323014
$316$	−14637.2	−2.60571
$317$	3563.21	0.631324	0.315662	−	0.948872i	$-0.397773\pi$
$317$	3563.21	0.631324	0.315662	+	0.948872i	$0.397773\pi$
$318$	−2330.75	−0.411012
$319$	0	0
$320$	3996.24	0.698114
$321$	−2393.93	−0.416250
$322$	−4639.44	−0.802938
$323$	−4562.76	−0.786003
$324$	−10040.7	−1.72166
$325$	−1784.53	−0.304578
$326$	11557.9	1.96360
$327$	3519.71	0.595231
$328$	−1741.26	−0.293126
$329$	6490.57	1.08765
$330$	0	0
$331$	1097.52	0.182251	0.0911253	−	0.995839i	$-0.470954\pi$
$331$	1097.52	0.182251	0.0911253	+	0.995839i	$0.470954\pi$
$332$	15061.5	2.48978
$333$	294.277	0.0484273
$334$	84.9958	0.0139244
$335$	−4696.88	−0.766025
$336$	4764.60	0.773602
$337$	−5297.23	−0.856257	−0.428129	−	0.903718i	$-0.640827\pi$
$337$	−5297.23	−0.856257	−0.428129	+	0.903718i	$0.640827\pi$
$338$	−12647.5	−2.03531
$339$	1489.49	0.238637
$340$	−2864.28	−0.456874
$341$	0	0
$342$	5725.95	0.905332
$343$	−2417.21	−0.380516
$344$	−1456.47	−0.228278
$345$	1421.54	0.221836
$346$	11638.9	1.80842
$347$	−5992.28	−0.927039	−0.463520	−	0.886087i	$-0.653414\pi$
$347$	−5992.28	−0.927039	−0.463520	+	0.886087i	$0.653414\pi$
$348$	11910.4	1.83467
$349$	6907.00	1.05938	0.529689	−	0.848192i	$-0.322309\pi$
$349$	6907.00	1.05938	0.529689	+	0.848192i	$0.322309\pi$
$350$	−2641.29	−0.403379
$351$	−5583.72	−0.849109
$352$	0	0
$353$	−7560.23	−1.13992	−0.569958	−	0.821674i	$-0.693040\pi$
$353$	−7560.23	−1.13992	−0.569958	+	0.821674i	$0.693040\pi$
$354$	−17574.9	−2.63869
$355$	−5423.10	−0.810784
$356$	7343.04	1.09320
$357$	−8131.41	−1.20549
$358$	399.876	0.0590338
$359$	−7797.45	−1.14633	−0.573167	−	0.819439i	$-0.694285\pi$
$359$	−7797.45	−1.14633	−0.573167	+	0.819439i	$0.694285\pi$
$360$	990.488	0.145009
$361$	877.426	0.127923
$362$	12004.0	1.74286
$363$	0	0
$364$	−19084.4	−2.74806
$365$	1697.36	0.243408
$366$	−7391.67	−1.05565
$367$	−2201.74	−0.313161	−0.156580	−	0.987665i	$-0.550047\pi$
$367$	−2201.74	−0.313161	−0.156580	+	0.987665i	$0.550047\pi$
$368$	−1334.78	−0.189076
$369$	1956.16	0.275972
$370$	430.413	0.0604760
$371$	−1997.27	−0.279496
$372$	−16076.5	−2.24067
$373$	−5084.28	−0.705775	−0.352888	−	0.935666i	$-0.614800\pi$
$373$	−5084.28	−0.705775	−0.352888	+	0.935666i	$0.614800\pi$
$374$	0	0
$375$	809.301	0.111446
$376$	−3559.96	−0.488273
$377$	11891.1	1.62447
$378$	−8264.49	−1.12455
$379$	−9019.04	−1.22237	−0.611183	−	0.791489i	$-0.709306\pi$
$379$	−9019.04	−1.22237	−0.611183	+	0.791489i	$0.709306\pi$
$380$	4856.54	0.655620
$381$	13274.4	1.78496
$382$	−10655.5	−1.42718
$383$	−8460.95	−1.12881	−0.564405	−	0.825498i	$-0.690894\pi$
$383$	−8460.95	−1.12881	−0.564405	+	0.825498i	$0.690894\pi$
$384$	−10208.6	−1.35666
$385$	0	0
$386$	−6269.03	−0.826646
$387$	1636.22	0.214919
$388$	−8560.69	−1.12011
$389$	−775.761	−0.101112	−0.0505561	−	0.998721i	$-0.516099\pi$
$389$	−775.761	−0.101112	−0.0505561	+	0.998721i	$0.516099\pi$
$390$	10083.7	1.30926
$391$	2277.97	0.294634
$392$	−3228.95	−0.416037
$393$	16086.2	2.06473
$394$	−9721.39	−1.24304
$395$	6627.36	0.844199
$396$	0	0
$397$	8216.54	1.03873	0.519366	−	0.854552i	$-0.326168\pi$
$397$	8216.54	1.03873	0.519366	+	0.854552i	$0.326168\pi$
$398$	−17913.6	−2.25609
$399$	13787.3	1.72989
$400$	−759.903	−0.0949879
$401$	4372.98	0.544579	0.272289	−	0.962215i	$-0.412219\pi$
$401$	4372.98	0.544579	0.272289	+	0.962215i	$0.412219\pi$
$402$	26540.4	3.29282
$403$	−16050.5	−1.98395
$404$	17368.4	2.13889
$405$	4546.20	0.557784
$406$	17600.1	2.15143
$407$	0	0
$408$	4459.92	0.541174
$409$	−1302.48	−0.157466	−0.0787328	−	0.996896i	$-0.525087\pi$
$409$	−1302.48	−0.157466	−0.0787328	+	0.996896i	$0.525087\pi$
$410$	2861.10	0.344633
$411$	350.777	0.0420987
$412$	−383.681	−0.0458802
$413$	−15060.3	−1.79436
$414$	−2858.69	−0.339365
$415$	−6819.47	−0.806637
$416$	−17051.3	−2.00963
$417$	−1480.64	−0.173878
$418$	0	0
$419$	5239.74	0.610926	0.305463	−	0.952204i	$-0.401189\pi$
$419$	5239.74	0.610926	0.305463	+	0.952204i	$0.401189\pi$
$420$	8654.96	1.00552
$421$	15874.3	1.83769	0.918845	−	0.394619i	$-0.129123\pi$
$421$	15874.3	1.83769	0.918845	+	0.394619i	$0.129123\pi$
$422$	17800.6	2.05336
$423$	3999.30	0.459699
$424$	1095.46	0.125473
$425$	1296.87	0.148018
$426$	30644.0	3.48523
$427$	−6334.07	−0.717862
$428$	4083.18	0.461141
$429$	0	0
$430$	2393.15	0.268391
$431$	−3523.78	−0.393816	−0.196908	−	0.980422i	$-0.563090\pi$
$431$	−3523.78	−0.393816	−0.196908	+	0.980422i	$0.563090\pi$
$432$	−2377.71	−0.264809
$433$	−16023.1	−1.77834	−0.889168	−	0.457581i	$-0.848716\pi$
$433$	−16023.1	−1.77834	−0.889168	+	0.457581i	$0.848716\pi$
$434$	−23756.4	−2.62752
$435$	−5392.75	−0.594397
$436$	−6003.36	−0.659424
$437$	−3862.43	−0.422803
$438$	−9591.16	−1.04631
$439$	3842.70	0.417773	0.208886	−	0.977940i	$-0.433016\pi$
$439$	3842.70	0.417773	0.208886	+	0.977940i	$0.433016\pi$
$440$	0	0
$441$	3627.45	0.391691
$442$	16158.8	1.73890
$443$	1996.23	0.214094	0.107047	−	0.994254i	$-0.465860\pi$
$443$	1996.23	0.214094	0.107047	+	0.994254i	$0.465860\pi$
$444$	−1410.38	−0.150751
$445$	−3324.75	−0.354176
$446$	775.860	0.0823722
$447$	7909.28	0.836904
$448$	−19350.4	−2.04067
$449$	8028.80	0.843881	0.421940	−	0.906624i	$-0.361349\pi$
$449$	8028.80	0.843881	0.421940	+	0.906624i	$0.361349\pi$
$450$	−1627.49	−0.170490
$451$	0	0
$452$	−2540.53	−0.264373
$453$	2854.64	0.296076
$454$	−4311.64	−0.445716
$455$	8640.95	0.890316
$456$	−7562.06	−0.776592
$457$	−1422.44	−0.145600	−0.0728000	−	0.997347i	$-0.523193\pi$
$457$	−1422.44	−0.145600	−0.0728000	+	0.997347i	$0.523193\pi$
$458$	2276.61	0.232268
$459$	4057.87	0.412647
$460$	−2424.64	−0.245760
$461$	−12510.0	−1.26388	−0.631939	−	0.775018i	$-0.717741\pi$
$461$	−12510.0	−1.26388	−0.631939	+	0.775018i	$0.717741\pi$
$462$	0	0
$463$	4643.04	0.466048	0.233024	−	0.972471i	$-0.425138\pi$
$463$	4643.04	0.466048	0.233024	+	0.972471i	$0.425138\pi$
$464$	5063.59	0.506619
$465$	7279.06	0.725932
$466$	−15134.5	−1.50449
$467$	10485.6	1.03900	0.519501	−	0.854470i	$-0.326118\pi$
$467$	10485.6	1.03900	0.519501	+	0.854470i	$0.326118\pi$
$468$	−11759.3	−1.16148
$469$	22743.0	2.23918
$470$	5849.42	0.574071
$471$	144.504	0.0141367
$472$	8260.29	0.805531
$473$	0	0
$474$	−37448.8	−3.62886
$475$	−2198.92	−0.212407
$476$	13869.2	1.33550
$477$	−1230.66	−0.118130
$478$	17494.4	1.67401
$479$	−12961.1	−1.23634	−0.618171	−	0.786043i	$-0.712126\pi$
$479$	−12961.1	−1.23634	−0.618171	+	0.786043i	$0.712126\pi$
$480$	7732.92	0.735329
$481$	−1408.09	−0.133479
$482$	7398.90	0.699193
$483$	−6883.32	−0.648451
$484$	0	0
$485$	3876.07	0.362894
$486$	−16472.3	−1.53745
$487$	3085.83	0.287130	0.143565	−	0.989641i	$-0.454143\pi$
$487$	3085.83	0.287130	0.143565	+	0.989641i	$0.454143\pi$
$488$	3474.12	0.322266
$489$	17147.9	1.58580
$490$	5305.54	0.489142
$491$	−7041.85	−0.647239	−0.323619	−	0.946187i	$-0.604900\pi$
$491$	−7041.85	−0.647239	−0.323619	+	0.946187i	$0.604900\pi$
$492$	−9375.23	−0.859082
$493$	−8641.68	−0.789455
$494$	−27398.2	−2.49535
$495$	0	0
$496$	−6834.76	−0.618730
$497$	26259.4	2.37001
$498$	38534.4	3.46740
$499$	−9776.64	−0.877079	−0.438539	−	0.898712i	$-0.644504\pi$
$499$	−9776.64	−0.877079	−0.438539	+	0.898712i	$0.644504\pi$
$500$	−1380.38	−0.123465
$501$	126.104	0.0112454
$502$	16376.6	1.45603
$503$	−2075.54	−0.183984	−0.0919920	−	0.995760i	$-0.529323\pi$
$503$	−2075.54	−0.183984	−0.0919920	+	0.995760i	$0.529323\pi$
$504$	−4796.09	−0.423878
$505$	−7863.98	−0.692956
$506$	0	0
$507$	−18764.6	−1.64372
$508$	−22641.4	−1.97746
$509$	9995.05	0.870379	0.435189	−	0.900339i	$-0.356681\pi$
$509$	9995.05	0.870379	0.435189	+	0.900339i	$0.356681\pi$
$510$	−7328.17	−0.636268
$511$	−8218.85	−0.711508
$512$	−10490.0	−0.905463
$513$	−6880.35	−0.592154
$514$	9103.03	0.781162
$515$	173.722	0.0148642
$516$	−7841.87	−0.669029
$517$	0	0
$518$	−2084.12	−0.176778
$519$	17268.2	1.46048
$520$	−4739.40	−0.399685
$521$	10207.7	0.858362	0.429181	−	0.903219i	$-0.358802\pi$
$521$	10207.7	0.858362	0.429181	+	0.903219i	$0.358802\pi$
$522$	10844.7	0.909309
$523$	15991.8	1.33704	0.668522	−	0.743692i	$-0.266927\pi$
$523$	15991.8	1.33704	0.668522	+	0.743692i	$0.266927\pi$
$524$	−27437.2	−2.28741
$525$	−3918.75	−0.325768
$526$	−123.887	−0.0102695
$527$	11664.4	0.964155
$528$	0	0
$529$	−10238.7	−0.841512
$530$	−1799.97	−0.147520
$531$	−9279.72	−0.758391
$532$	−23516.1	−1.91645
$533$	−9360.06	−0.760655
$534$	18787.0	1.52246
$535$	−1848.77	−0.149400
$536$	−12474.1	−1.00522
$537$	593.277	0.0476756
$538$	−4835.54	−0.387500
$539$	0	0
$540$	−4319.14	−0.344197
$541$	4784.39	0.380216	0.190108	−	0.981763i	$-0.439116\pi$
$541$	4784.39	0.380216	0.190108	+	0.981763i	$0.439116\pi$
$542$	−13123.8	−1.04006
$543$	17809.7	1.40753
$544$	12391.7	0.976637
$545$	2718.17	0.213640
$546$	−48826.9	−3.82710
$547$	−18081.6	−1.41337	−0.706686	−	0.707527i	$-0.749811\pi$
$547$	−18081.6	−1.41337	−0.706686	+	0.707527i	$0.749811\pi$
$548$	−598.300	−0.0466389
$549$	−3902.87	−0.303407
$550$	0	0
$551$	14652.5	1.13288
$552$	3775.37	0.291106
$553$	−32090.6	−2.46769
$554$	33578.4	2.57510
$555$	638.583	0.0488403
$556$	2525.43	0.192630
$557$	5283.98	0.401956	0.200978	−	0.979596i	$-0.435588\pi$
$557$	5283.98	0.401956	0.200978	+	0.979596i	$0.435588\pi$
$558$	−14638.0	−1.11053
$559$	−7829.17	−0.592377
$560$	3679.56	0.277661
$561$	0	0
$562$	17596.6	1.32076
$563$	−1802.78	−0.134952	−0.0674762	−	0.997721i	$-0.521495\pi$
$563$	−1802.78	−0.134952	−0.0674762	+	0.997721i	$0.521495\pi$
$564$	−19167.3	−1.43101
$565$	1150.29	0.0856514
$566$	−6382.81	−0.474010
$567$	−22013.4	−1.63047
$568$	−14402.8	−1.06396
$569$	−12941.8	−0.953516	−0.476758	−	0.879035i	$-0.658188\pi$
$569$	−12941.8	−0.953516	−0.476758	+	0.879035i	$0.658188\pi$
$570$	12425.3	0.913052
$571$	−10130.7	−0.742479	−0.371240	−	0.928537i	$-0.621067\pi$
$571$	−10130.7	−0.742479	−0.371240	+	0.928537i	$0.621067\pi$
$572$	0	0
$573$	−15809.0	−1.15259
$574$	−13853.9	−1.00740
$575$	1097.82	0.0796211
$576$	−11923.1	−0.862496
$577$	22781.5	1.64369	0.821843	−	0.569714i	$-0.192946\pi$
$577$	22781.5	1.64369	0.821843	+	0.569714i	$0.192946\pi$
$578$	9696.38	0.697779
$579$	−9301.06	−0.667598
$580$	9198.09	0.658499
$581$	33020.9	2.35789
$582$	−21902.3	−1.55993
$583$	0	0
$584$	4507.89	0.319414
$585$	5324.30	0.376296
$586$	22989.8	1.62065
$587$	−22791.8	−1.60258	−0.801292	−	0.598273i	$-0.795854\pi$
$587$	−22791.8	−1.60258	−0.801292	+	0.598273i	$0.795854\pi$
$588$	−17385.2	−1.21931
$589$	−19777.7	−1.38357
$590$	−13572.6	−0.947077
$591$	−14423.2	−1.00387
$592$	−599.606	−0.0416278
$593$	5399.20	0.373893	0.186947	−	0.982370i	$-0.440141\pi$
$593$	5399.20	0.373893	0.186947	+	0.982370i	$0.440141\pi$
$594$	0	0
$595$	−6279.65	−0.432673
$596$	−13490.4	−0.927160
$597$	−26577.5	−1.82202
$598$	13678.6	0.935383
$599$	−20755.0	−1.41574	−0.707869	−	0.706343i	$-0.750343\pi$
$599$	−20755.0	−1.41574	−0.707869	+	0.706343i	$0.750343\pi$
$600$	2149.36	0.146246
$601$	−13439.7	−0.912171	−0.456086	−	0.889936i	$-0.650749\pi$
$601$	−13439.7	−0.912171	−0.456086	+	0.889936i	$0.650749\pi$
$602$	−11588.0	−0.784537
$603$	14013.6	0.946397
$604$	−4868.98	−0.328006
$605$	0	0
$606$	44436.6	2.97873
$607$	−4140.82	−0.276888	−0.138444	−	0.990370i	$-0.544210\pi$
$607$	−4140.82	−0.276888	−0.138444	+	0.990370i	$0.544210\pi$
$608$	−21010.9	−1.40148
$609$	26112.5	1.73749
$610$	−5708.38	−0.378894
$611$	−19136.3	−1.26706
$612$	8545.83	0.564452
$613$	11697.3	0.770714	0.385357	−	0.922768i	$-0.374078\pi$
$613$	11697.3	0.770714	0.385357	+	0.922768i	$0.374078\pi$
$614$	24916.4	1.63769
$615$	4244.88	0.278325
$616$	0	0
$617$	2547.40	0.166215	0.0831073	−	0.996541i	$-0.473516\pi$
$617$	2547.40	0.166215	0.0831073	+	0.996541i	$0.473516\pi$
$618$	−981.638	−0.0638953
$619$	12888.8	0.836903	0.418451	−	0.908239i	$-0.362573\pi$
$619$	12888.8	0.836903	0.418451	+	0.908239i	$0.362573\pi$
$620$	−12415.4	−0.804220
$621$	3435.03	0.221969
$622$	14459.5	0.932112
$623$	16098.9	1.03530
$624$	−14047.6	−0.901208
$625$	625.000	0.0400000
$626$	27898.7	1.78124
$627$	0	0
$628$	−246.471	−0.0156612
$629$	1023.31	0.0648678
$630$	7880.52	0.498361
$631$	−26494.7	−1.67154	−0.835768	−	0.549083i	$-0.814977\pi$
$631$	−26494.7	−1.67154	−0.835768	+	0.549083i	$0.814977\pi$
$632$	17601.1	1.10781
$633$	26409.9	1.65829
$634$	−15549.2	−0.974037
$635$	10251.5	0.640657
$636$	5898.14	0.367730
$637$	−17357.0	−1.07961
$638$	0	0
$639$	16180.3	1.00170
$640$	−7883.83	−0.486931
$641$	−2538.72	−0.156433	−0.0782163	−	0.996936i	$-0.524922\pi$
$641$	−2538.72	−0.156433	−0.0782163	+	0.996936i	$0.524922\pi$
$642$	10446.7	0.642210
$643$	27573.2	1.69111	0.845554	−	0.533890i	$-0.179270\pi$
$643$	27573.2	1.69111	0.845554	+	0.533890i	$0.179270\pi$
$644$	11740.5	0.718383
$645$	3550.60	0.216752
$646$	19911.1	1.21268
$647$	−9918.47	−0.602682	−0.301341	−	0.953516i	$-0.597434\pi$
$647$	−9918.47	−0.602682	−0.301341	+	0.953516i	$0.597434\pi$
$648$	12073.9	0.731957
$649$	0	0
$650$	7787.38	0.469917
$651$	−35246.3	−2.12198
$652$	−29248.1	−1.75682
$653$	5766.21	0.345558	0.172779	−	0.984961i	$-0.444725\pi$
$653$	5766.21	0.345558	0.172779	+	0.984961i	$0.444725\pi$
$654$	−15359.4	−0.918350
$655$	12422.9	0.741073
$656$	−3985.78	−0.237223
$657$	−5064.22	−0.300722
$658$	−28323.7	−1.67808
$659$	2659.35	0.157198	0.0785991	−	0.996906i	$-0.474955\pi$
$659$	2659.35	0.157198	0.0785991	+	0.996906i	$0.474955\pi$
$660$	0	0
$661$	2724.52	0.160320	0.0801600	−	0.996782i	$-0.474457\pi$
$661$	2724.52	0.160320	0.0801600	+	0.996782i	$0.474457\pi$
$662$	−4789.37	−0.281185
$663$	23974.0	1.40434
$664$	−18111.3	−1.05852
$665$	10647.5	0.620891
$666$	−1284.18	−0.0747159
$667$	−7315.27	−0.424660
$668$	−215.088	−0.0124581
$669$	1151.11	0.0665237
$670$	20496.4	1.18186
$671$	0	0
$672$	−37444.0	−2.14945
$673$	22661.2	1.29796	0.648980	−	0.760806i	$-0.275196\pi$
$673$	22661.2	1.29796	0.648980	+	0.760806i	$0.275196\pi$
$674$	23116.2	1.32107
$675$	1955.60	0.111513
$676$	32005.6	1.82098
$677$	30128.8	1.71041	0.855203	−	0.518294i	$-0.173433\pi$
$677$	30128.8	1.71041	0.855203	+	0.518294i	$0.173433\pi$
$678$	−6499.87	−0.368180
$679$	−18768.5	−1.06078
$680$	3444.27	0.194238
$681$	−6396.98	−0.359960
$682$	0	0
$683$	−30674.6	−1.71849	−0.859247	−	0.511561i	$-0.829067\pi$
$683$	−30674.6	−1.71849	−0.859247	+	0.511561i	$0.829067\pi$
$684$	−14489.9	−0.809995
$685$	270.895	0.0151100
$686$	10548.3	0.587077
$687$	3377.69	0.187579
$688$	−3333.89	−0.184743
$689$	5888.59	0.325599
$690$	−6203.37	−0.342258
$691$	16570.2	0.912242	0.456121	−	0.889918i	$-0.349239\pi$
$691$	16570.2	0.912242	0.456121	+	0.889918i	$0.349239\pi$
$692$	−29453.2	−1.61798
$693$	0	0
$694$	26149.3	1.43028
$695$	−1143.45	−0.0624081
$696$	−14322.2	−0.780003
$697$	6802.25	0.369661
$698$	−30140.9	−1.63446
$699$	−22454.3	−1.21502
$700$	6683.98	0.360901
$701$	4072.03	0.219398	0.109699	−	0.993965i	$-0.465011\pi$
$701$	4072.03	0.219398	0.109699	+	0.993965i	$0.465011\pi$
$702$	24366.4	1.31004
$703$	−1735.07	−0.0930861
$704$	0	0
$705$	8678.51	0.463619
$706$	32991.5	1.75871
$707$	38078.6	2.02559
$708$	44474.7	2.36082
$709$	1994.70	0.105659	0.0528296	−	0.998604i	$-0.483176\pi$
$709$	1994.70	0.105659	0.0528296	+	0.998604i	$0.483176\pi$
$710$	23665.5	1.25091
$711$	−19773.3	−1.04298
$712$	−8829.96	−0.464771
$713$	9874.04	0.518634
$714$	35484.1	1.85988
$715$	0	0
$716$	−1011.92	−0.0528172
$717$	25955.7	1.35193
$718$	34026.7	1.76862
$719$	22520.0	1.16809	0.584044	−	0.811722i	$-0.301469\pi$
$719$	22520.0	1.16809	0.584044	+	0.811722i	$0.301469\pi$
$720$	2267.24	0.117354
$721$	−841.185	−0.0434499
$722$	−3828.94	−0.197366
$723$	10977.4	0.564667
$724$	−30377.0	−1.55932
$725$	−4164.67	−0.213341
$726$	0	0
$727$	−33707.0	−1.71956	−0.859782	−	0.510661i	$-0.829401\pi$
$727$	−33707.0	−1.71956	−0.859782	+	0.510661i	$0.829401\pi$
$728$	22948.9	1.16833
$729$	110.279	0.00560275
$730$	−7406.98	−0.375540
$731$	5689.71	0.287882
$732$	18705.2	0.944486
$733$	26949.9	1.35800	0.679002	−	0.734136i	$-0.262413\pi$
$733$	26949.9	1.35800	0.679002	+	0.734136i	$0.262413\pi$
$734$	9608.02	0.483159
$735$	7871.58	0.395031
$736$	10489.7	0.525348
$737$	0	0
$738$	−8536.34	−0.425782
$739$	19223.0	0.956872	0.478436	−	0.878122i	$-0.341204\pi$
$739$	19223.0	0.956872	0.478436	+	0.878122i	$0.341204\pi$
$740$	−1089.19	−0.0541075
$741$	−40649.4	−2.01524
$742$	8715.72	0.431219
$743$	15273.3	0.754136	0.377068	−	0.926186i	$-0.376932\pi$
$743$	15273.3	0.754136	0.377068	+	0.926186i	$0.376932\pi$
$744$	19331.9	0.952611
$745$	6108.11	0.300381
$746$	22186.9	1.08890
$747$	20346.5	0.996573
$748$	0	0
$749$	8951.99	0.436714
$750$	−3531.65	−0.171944
$751$	−14085.8	−0.684420	−0.342210	−	0.939624i	$-0.611175\pi$
$751$	−14085.8	−0.684420	−0.342210	+	0.939624i	$0.611175\pi$
$752$	−8148.79	−0.395154
$753$	24297.2	1.17588
$754$	−51890.9	−2.50631
$755$	2204.55	0.106267
$756$	20913.9	1.00613
$757$	−31013.0	−1.48902	−0.744510	−	0.667611i	$-0.767317\pi$
$757$	−31013.0	−1.48902	−0.744510	+	0.667611i	$0.767317\pi$
$758$	39357.5	1.88592
$759$	0	0
$760$	−5839.96	−0.278734
$761$	25582.1	1.21859	0.609297	−	0.792942i	$-0.291452\pi$
$761$	25582.1	1.21859	0.609297	+	0.792942i	$0.291452\pi$
$762$	−57927.4	−2.75392
$763$	−13161.8	−0.624494
$764$	26964.5	1.27689
$765$	−3869.34	−0.182871
$766$	36922.1	1.74158
$767$	44402.7	2.09034
$768$	3151.47	0.148072
$769$	23181.0	1.08703	0.543517	−	0.839398i	$-0.317092\pi$
$769$	23181.0	1.08703	0.543517	+	0.839398i	$0.317092\pi$
$770$	0	0
$771$	13505.7	0.630865
$772$	15864.3	0.739595
$773$	3409.67	0.158651	0.0793256	−	0.996849i	$-0.474723\pi$
$773$	3409.67	0.158651	0.0793256	+	0.996849i	$0.474723\pi$
$774$	−7140.18	−0.331587
$775$	5621.41	0.260551
$776$	10294.2	0.476211
$777$	−3092.12	−0.142766
$778$	3385.29	0.156001
$779$	−11533.6	−0.530468
$780$	−25517.7	−1.17138
$781$	0	0
$782$	−9940.66	−0.454575
$783$	−13031.1	−0.594755
$784$	−7391.12	−0.336694
$785$	111.596	0.00507392
$786$	−70197.4	−3.18557
$787$	−18292.0	−0.828510	−0.414255	−	0.910161i	$-0.635958\pi$
$787$	−18292.0	−0.828510	−0.414255	+	0.910161i	$0.635958\pi$
$788$	24600.7	1.11214
$789$	−183.805	−0.00829359
$790$	−28920.6	−1.30247
$791$	−5569.87	−0.250369
$792$	0	0
$793$	18674.9	0.836274
$794$	−35855.6	−1.60260
$795$	−2670.53	−0.119137
$796$	45331.6	2.01851
$797$	−32329.6	−1.43685	−0.718427	−	0.695603i	$-0.755137\pi$
$797$	−32329.6	−1.43685	−0.718427	+	0.695603i	$0.755137\pi$
$798$	−60165.3	−2.66896
$799$	13907.0	0.615761
$800$	5971.92	0.263924
$801$	9919.70	0.437572
$802$	−19082.9	−0.840202
$803$	0	0
$804$	−67162.5	−2.94607
$805$	−5315.79	−0.232742
$806$	70041.6	3.06093
$807$	−7174.26	−0.312944
$808$	−20885.4	−0.909338
$809$	22288.6	0.968633	0.484316	−	0.874893i	$-0.339068\pi$
$809$	22288.6	0.968633	0.484316	+	0.874893i	$0.339068\pi$
$810$	−19838.8	−0.860574
$811$	526.439	0.0227938	0.0113969	−	0.999935i	$-0.496372\pi$
$811$	526.439	0.0227938	0.0113969	+	0.999935i	$0.496372\pi$
$812$	−44538.5	−1.92487
$813$	−19471.1	−0.839952
$814$	0	0
$815$	13242.8	0.569174
$816$	10208.8	0.437966
$817$	−9647.23	−0.413113
$818$	5683.79	0.242945
$819$	−25781.1	−1.09996
$820$	−7240.22	−0.308341
$821$	5254.99	0.223386	0.111693	−	0.993743i	$-0.464373\pi$
$821$	5254.99	0.223386	0.111693	+	0.993743i	$0.464373\pi$
$822$	−1530.73	−0.0649519
$823$	17504.7	0.741406	0.370703	−	0.928752i	$-0.379117\pi$
$823$	17504.7	0.741406	0.370703	+	0.928752i	$0.379117\pi$
$824$	461.374	0.0195057
$825$	0	0
$826$	65720.6	2.76842
$827$	11703.4	0.492100	0.246050	−	0.969257i	$-0.420867\pi$
$827$	11703.4	0.492100	0.246050	+	0.969257i	$0.420867\pi$
$828$	7234.13	0.303627
$829$	14165.4	0.593467	0.296734	−	0.954960i	$-0.404103\pi$
$829$	14165.4	0.593467	0.296734	+	0.954960i	$0.404103\pi$
$830$	29759.0	1.24452
$831$	49818.6	2.07965
$832$	57051.2	2.37728
$833$	12613.9	0.524665
$834$	6461.24	0.268267
$835$	97.3867	0.00403617
$836$	0	0
$837$	17589.2	0.726369
$838$	−22865.3	−0.942565
$839$	−38871.3	−1.59951	−0.799754	−	0.600328i	$-0.795037\pi$
$839$	−38871.3	−1.59951	−0.799754	+	0.600328i	$0.795037\pi$
$840$	−10407.5	−0.427493
$841$	3362.13	0.137854
$842$	−69272.8	−2.83527
$843$	26107.3	1.06665
$844$	−45045.7	−1.83713
$845$	−14491.3	−0.589961
$846$	−17452.3	−0.709245
$847$	0	0
$848$	2507.53	0.101544
$849$	−9469.88	−0.382810
$850$	−5659.33	−0.228369
$851$	866.239	0.0348934
$852$	−77546.9	−3.11821
$853$	28088.7	1.12748	0.563739	−	0.825953i	$-0.309362\pi$
$853$	28088.7	1.12748	0.563739	+	0.825953i	$0.309362\pi$
$854$	27640.8	1.10755
$855$	6560.69	0.262422
$856$	−4910.00	−0.196052
$857$	30501.3	1.21576	0.607879	−	0.794030i	$-0.292021\pi$
$857$	30501.3	1.21576	0.607879	+	0.794030i	$0.292021\pi$
$858$	0	0
$859$	28243.1	1.12182	0.560909	−	0.827877i	$-0.310452\pi$
$859$	28243.1	1.12182	0.560909	+	0.827877i	$0.310452\pi$
$860$	−6056.05	−0.240127
$861$	−20554.3	−0.813576
$862$	15377.2	0.607598
$863$	−23998.9	−0.946618	−0.473309	−	0.880897i	$-0.656941\pi$
$863$	−23998.9	−0.946618	−0.473309	+	0.880897i	$0.656941\pi$
$864$	18685.9	0.735772
$865$	13335.7	0.524194
$866$	69921.9	2.74370
$867$	14386.1	0.563525
$868$	60117.4	2.35083
$869$	0	0
$870$	23533.0	0.917063
$871$	−67053.8	−2.60853
$872$	7219.00	0.280351
$873$	−11564.6	−0.448342
$874$	16855.0	0.652320
$875$	−3026.34	−0.116925
$876$	24271.2	0.936126
$877$	−14032.4	−0.540295	−0.270148	−	0.962819i	$-0.587073\pi$
$877$	−14032.4	−0.540295	−0.270148	+	0.962819i	$0.587073\pi$
$878$	−16768.9	−0.644559
$879$	34108.9	1.30884
$880$	0	0
$881$	−28208.8	−1.07875	−0.539375	−	0.842065i	$-0.681340\pi$
$881$	−28208.8	−1.07875	−0.539375	+	0.842065i	$0.681340\pi$
$882$	−15829.6	−0.604318
$883$	4261.18	0.162401	0.0812005	−	0.996698i	$-0.474125\pi$
$883$	4261.18	0.162401	0.0812005	+	0.996698i	$0.474125\pi$
$884$	−40891.1	−1.55579
$885$	−20137.0	−0.764858
$886$	−8711.20	−0.330314
$887$	−19077.9	−0.722181	−0.361090	−	0.932531i	$-0.617595\pi$
$887$	−19077.9	−0.722181	−0.361090	+	0.932531i	$0.617595\pi$
$888$	1695.97	0.0640911
$889$	−49639.1	−1.87271
$890$	14508.6	0.546439
$891$	0	0
$892$	−1963.37	−0.0736979
$893$	−23580.1	−0.883625
$894$	−34514.8	−1.29121
$895$	458.171	0.0171117
$896$	38174.7	1.42336
$897$	20294.3	0.755414
$898$	−35036.3	−1.30198
$899$	−37458.1	−1.38965
$900$	4118.48	0.152536
$901$	−4279.43	−0.158233
$902$	0	0
$903$	−17192.6	−0.633591
$904$	3054.97	0.112397
$905$	13753.9	0.505190
$906$	−12457.1	−0.456800
$907$	15949.0	0.583877	0.291939	−	0.956437i	$-0.405700\pi$
$907$	15949.0	0.583877	0.291939	+	0.956437i	$0.405700\pi$
$908$	10910.9	0.398780
$909$	23462.9	0.856123
$910$	−37707.6	−1.37362
$911$	−32175.7	−1.17017	−0.585087	−	0.810970i	$-0.698940\pi$
$911$	−32175.7	−1.17017	−0.585087	+	0.810970i	$0.698940\pi$
$912$	−17309.7	−0.628487
$913$	0	0
$914$	6207.31	0.224638
$915$	−8469.25	−0.305994
$916$	−5761.13	−0.207809
$917$	−60153.5	−2.16624
$918$	−17707.8	−0.636652
$919$	−39962.4	−1.43443	−0.717213	−	0.696854i	$-0.754582\pi$
$919$	−39962.4	−1.43443	−0.717213	+	0.696854i	$0.754582\pi$
$920$	2915.61	0.104484
$921$	36967.3	1.32260
$922$	54591.4	1.94997
$923$	−77421.4	−2.76095
$924$	0	0
$925$	493.160	0.0175297
$926$	−20261.4	−0.719040
$927$	−518.314	−0.0183643
$928$	−39793.7	−1.40764
$929$	3106.30	0.109703	0.0548517	−	0.998495i	$-0.482531\pi$
$929$	3106.30	0.109703	0.0548517	+	0.998495i	$0.482531\pi$
$930$	−31764.6	−1.12000
$931$	−21387.6	−0.752900
$932$	38298.9	1.34605
$933$	21452.9	0.752772
$934$	−45757.2	−1.60302
$935$	0	0
$936$	14140.4	0.493798
$937$	−3732.34	−0.130128	−0.0650641	−	0.997881i	$-0.520725\pi$
$937$	−3732.34	−0.130128	−0.0650641	+	0.997881i	$0.520725\pi$
$938$	−99246.6	−3.45471
$939$	41392.1	1.43853
$940$	−14802.4	−0.513618
$941$	8847.41	0.306501	0.153250	−	0.988187i	$-0.451026\pi$
$941$	8847.41	0.306501	0.153250	+	0.988187i	$0.451026\pi$
$942$	−630.589	−0.0218107
$943$	5758.18	0.198846
$944$	18907.9	0.651908
$945$	−9469.31	−0.325965
$946$	0	0
$947$	30503.7	1.04671	0.523356	−	0.852114i	$-0.324680\pi$
$947$	30503.7	1.04671	0.523356	+	0.852114i	$0.324680\pi$
$948$	94767.1	3.24672
$949$	24231.9	0.828872
$950$	9595.73	0.327712
$951$	−23069.7	−0.786630
$952$	−16677.7	−0.567780
$953$	3252.13	0.110542	0.0552711	−	0.998471i	$-0.482398\pi$
$953$	3252.13	0.110542	0.0552711	+	0.998471i	$0.482398\pi$
$954$	5370.38	0.182256
$955$	−12208.9	−0.413685
$956$	−44271.0	−1.49773
$957$	0	0
$958$	56560.0	1.90749
$959$	−1311.72	−0.0441684
$960$	−25873.3	−0.869850
$961$	20769.4	0.697169
$962$	6144.67	0.205938
$963$	5515.96	0.184579
$964$	−18723.5	−0.625563
$965$	−7182.95	−0.239614
$966$	30037.6	1.00046
$967$	30327.7	1.00855	0.504277	−	0.863542i	$-0.331759\pi$
$967$	30327.7	1.00855	0.504277	+	0.863542i	$0.331759\pi$
$968$	0	0
$969$	29541.2	0.979360
$970$	−16914.5	−0.559889
$971$	−16791.6	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$971$	−16791.6	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$972$	41684.4	1.37554
$973$	5536.76	0.182426
$974$	−13466.0	−0.442997
$975$	11553.8	0.379504
$976$	7952.31	0.260807
$977$	−42016.4	−1.37587	−0.687934	−	0.725773i	$-0.741482\pi$
$977$	−42016.4	−1.37587	−0.687934	+	0.725773i	$0.741482\pi$
$978$	−74830.5	−2.44664
$979$	0	0
$980$	−13426.1	−0.437633
$981$	−8109.92	−0.263945
$982$	30729.4	0.998590
$983$	19773.6	0.641586	0.320793	−	0.947149i	$-0.396051\pi$
$983$	19773.6	0.641586	0.320793	+	0.947149i	$0.396051\pi$
$984$	11273.7	0.365235
$985$	−11138.6	−0.360310
$986$	37710.8	1.21801
$987$	−42022.6	−1.35521
$988$	69333.1	2.23257
$989$	4816.40	0.154856
$990$	0	0
$991$	−29858.0	−0.957085	−0.478542	−	0.878064i	$-0.658835\pi$
$991$	−29858.0	−0.957085	−0.478542	+	0.878064i	$0.658835\pi$
$992$	53712.9	1.71914
$993$	−7105.77	−0.227084
$994$	−114592.	−3.65657
$995$	−20525.0	−0.653957
$996$	−97514.2	−3.10226
$997$	5183.46	0.164656	0.0823278	−	0.996605i	$-0.473765\pi$
$997$	5183.46	0.164656	0.0823278	+	0.996605i	$0.473765\pi$
$998$	42663.6	1.35320
$999$	1543.08	0.0488697

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.4.a.q.1.2		12
11.2	odd	10		55.4.g.a.26.1	✓	24
11.6	odd	10		55.4.g.a.36.1	yes	24
11.10	odd	2		605.4.a.s.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.g.a.26.1	✓	24	11.2	odd	10
55.4.g.a.36.1	yes	24	11.6	odd	10
605.4.a.q.1.2		12	1.1	even	1	trivial
605.4.a.s.1.11		12	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	605.a (trivial)

\(f(q)\)	\(=\)	\(q-4.36383 q^{2} -6.47441 q^{3} +11.0430 q^{4} -5.00000 q^{5} +28.2532 q^{6} +24.2107 q^{7} -13.2791 q^{8} +14.9180 q^{9} +21.8191 q^{10} -71.4969 q^{12} -71.3811 q^{13} -105.652 q^{14} +32.3720 q^{15} -30.3961 q^{16} +51.8749 q^{17} -65.0994 q^{18} -87.9570 q^{19} -55.2150 q^{20} -156.750 q^{21} +43.9127 q^{23} +85.9745 q^{24} +25.0000 q^{25} +311.495 q^{26} +78.2241 q^{27} +267.359 q^{28} -166.587 q^{29} -141.266 q^{30} +224.856 q^{31} +238.877 q^{32} -226.373 q^{34} -121.054 q^{35} +164.739 q^{36} +19.7264 q^{37} +383.829 q^{38} +462.151 q^{39} +66.3957 q^{40} +131.128 q^{41} +684.031 q^{42} +109.681 q^{43} -74.5898 q^{45} -191.627 q^{46} +268.086 q^{47} +196.797 q^{48} +243.160 q^{49} -109.096 q^{50} -335.860 q^{51} -788.262 q^{52} -82.4950 q^{53} -341.356 q^{54} -321.498 q^{56} +569.469 q^{57} +726.956 q^{58} -622.051 q^{59} +357.484 q^{60} -261.622 q^{61} -981.234 q^{62} +361.175 q^{63} -799.248 q^{64} +356.906 q^{65} +939.377 q^{67} +572.855 q^{68} -284.309 q^{69} +528.258 q^{70} +1084.62 q^{71} -198.098 q^{72} -339.471 q^{73} -86.0826 q^{74} -161.860 q^{75} -971.309 q^{76} -2016.75 q^{78} -1325.47 q^{79} +151.981 q^{80} -909.239 q^{81} -572.219 q^{82} +1363.89 q^{83} -1730.99 q^{84} -259.375 q^{85} -478.630 q^{86} +1078.55 q^{87} +664.950 q^{89} +325.497 q^{90} -1728.19 q^{91} +484.928 q^{92} -1455.81 q^{93} -1169.88 q^{94} +439.785 q^{95} -1546.58 q^{96} -775.214 q^{97} -1061.11 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + q^{3} + 39 q^{4} - 60 q^{5} - 10 q^{6} - 41 q^{7} + 9 q^{8} + 131 q^{9} + 5 q^{10} - 29 q^{12} - 109 q^{13} - 247 q^{14} - 5 q^{15} + 283 q^{16} + 167 q^{17} - 135 q^{18} - 332 q^{19} - 195 q^{20}+ \cdots + 4209 q^{98}+O(q^{100}) \) 12 * q - q^2 + q^3 + 39 * q^4 - 60 * q^5 - 10 * q^6 - 41 * q^7 + 9 * q^8 + 131 * q^9 + 5 * q^10 - 29 * q^12 - 109 * q^13 - 247 * q^14 - 5 * q^15 + 283 * q^16 + 167 * q^17 - 135 * q^18 - 332 * q^19 - 195 * q^20 - 204 * q^21 - 193 * q^23 - 444 * q^24 + 300 * q^25 + 546 * q^26 + 334 * q^27 - 244 * q^28 - 216 * q^29 + 50 * q^30 - 252 * q^31 - 234 * q^32 - 665 * q^34 + 205 * q^35 - 473 * q^36 - 45 * q^37 - 238 * q^38 - 1144 * q^39 - 45 * q^40 - 998 * q^41 - 248 * q^42 - 371 * q^43 - 655 * q^45 - 2567 * q^46 + 257 * q^47 + 1605 * q^48 + 1057 * q^49 - 25 * q^50 - 2064 * q^51 - 3488 * q^52 - 201 * q^53 - 1140 * q^54 - 2710 * q^56 - 294 * q^57 + 1090 * q^58 + 20 * q^59 + 145 * q^60 - 1662 * q^61 - 737 * q^62 - 1911 * q^63 + 527 * q^64 + 545 * q^65 - 1941 * q^67 + 1468 * q^68 + 2178 * q^69 + 1235 * q^70 + 1032 * q^71 - 3514 * q^72 - 2441 * q^73 - 2697 * q^74 + 25 * q^75 - 3852 * q^76 - 3166 * q^78 - 2406 * q^79 - 1415 * q^80 - 204 * q^81 + 2956 * q^82 + 953 * q^83 - 4269 * q^84 - 835 * q^85 - 692 * q^86 - 910 * q^87 - 4204 * q^89 + 675 * q^90 - 328 * q^91 + 6565 * q^92 + 1630 * q^93 - 1196 * q^94 + 1660 * q^95 - 6253 * q^96 + 543 * q^97 + 4209 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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