LMFDB - Embedded newform 239.4.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [239,4,Mod(1,239)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(239, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("239.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	239.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:	$14.1014564914$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.43192 q^{2} +6.11724 q^{3} +11.6419 q^{4} -11.7413 q^{5} -27.1111 q^{6} +6.42321 q^{7} -16.1407 q^{8} +10.4206 q^{9} +O(q^{10})$	\(q-4.43192 q^{2} +6.11724 q^{3} +11.6419 q^{4} -11.7413 q^{5} -27.1111 q^{6} +6.42321 q^{7} -16.1407 q^{8} +10.4206 q^{9} +52.0364 q^{10} +33.6179 q^{11} +71.2164 q^{12} -66.2278 q^{13} -28.4672 q^{14} -71.8242 q^{15} -21.6011 q^{16} -29.6611 q^{17} -46.1834 q^{18} -27.1748 q^{19} -136.691 q^{20} +39.2923 q^{21} -148.992 q^{22} +186.640 q^{23} -98.7365 q^{24} +12.8574 q^{25} +293.516 q^{26} -101.420 q^{27} +74.7785 q^{28} +21.8413 q^{29} +318.319 q^{30} -193.238 q^{31} +224.860 q^{32} +205.649 q^{33} +131.456 q^{34} -75.4166 q^{35} +121.316 q^{36} -67.8794 q^{37} +120.437 q^{38} -405.131 q^{39} +189.512 q^{40} -74.4545 q^{41} -174.140 q^{42} -324.175 q^{43} +391.377 q^{44} -122.351 q^{45} -827.173 q^{46} -202.273 q^{47} -132.139 q^{48} -301.742 q^{49} -56.9832 q^{50} -181.444 q^{51} -771.019 q^{52} -254.801 q^{53} +449.485 q^{54} -394.717 q^{55} -103.675 q^{56} -166.235 q^{57} -96.7990 q^{58} -474.685 q^{59} -836.171 q^{60} +179.092 q^{61} +856.414 q^{62} +66.9339 q^{63} -823.752 q^{64} +777.599 q^{65} -911.420 q^{66} +294.098 q^{67} -345.313 q^{68} +1141.72 q^{69} +334.241 q^{70} -126.860 q^{71} -168.196 q^{72} -920.278 q^{73} +300.836 q^{74} +78.6521 q^{75} -316.367 q^{76} +215.935 q^{77} +1795.51 q^{78} +31.6274 q^{79} +253.624 q^{80} -901.767 q^{81} +329.977 q^{82} -1354.73 q^{83} +457.438 q^{84} +348.259 q^{85} +1436.72 q^{86} +133.609 q^{87} -542.617 q^{88} -669.715 q^{89} +542.252 q^{90} -425.395 q^{91} +2172.85 q^{92} -1182.08 q^{93} +896.459 q^{94} +319.067 q^{95} +1375.52 q^{96} +1611.14 q^{97} +1337.30 q^{98} +350.320 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 4 q^{2} - 13 q^{3} + 50 q^{4} - 37 q^{5} - 42 q^{6} - 52 q^{7} - 69 q^{8} + 69 q^{9}+O(q^{10}) $ 22 * q - 4 * q^2 - 13 * q^3 + 50 * q^4 - 37 * q^5 - 42 * q^6 - 52 * q^7 - 69 * q^8 + 69 * q^9	\( 22 q - 4 q^{2} - 13 q^{3} + 50 q^{4} - 37 q^{5} - 42 q^{6} - 52 q^{7} - 69 q^{8} + 69 q^{9} - 93 q^{10} - 77 q^{11} - 152 q^{12} - 218 q^{13} - 111 q^{14} - 142 q^{15} - 42 q^{16} - 219 q^{17} - 291 q^{18} - 476 q^{19} - 314 q^{20} - 474 q^{21} - 390 q^{22} - 202 q^{23} - 497 q^{24} - 271 q^{25} - 220 q^{26} - 244 q^{27} - 515 q^{28} - 307 q^{29} - 303 q^{30} - 1001 q^{31} - 771 q^{32} - 984 q^{33} - 1297 q^{34} - 430 q^{35} - 616 q^{36} - 922 q^{37} + 49 q^{38} - 542 q^{39} - 1344 q^{40} - 1188 q^{41} + 58 q^{42} - 192 q^{43} - 547 q^{44} - 1569 q^{45} - 1178 q^{46} - 102 q^{47} - 1209 q^{48} - 1952 q^{49} - 471 q^{50} - 834 q^{51} - 1785 q^{52} - 580 q^{53} - 1272 q^{54} - 1730 q^{55} - 804 q^{56} - 806 q^{57} - 1156 q^{58} - 1528 q^{59} + 489 q^{60} - 1631 q^{61} + 2206 q^{62} - 318 q^{63} + 327 q^{64} + 44 q^{65} + 3267 q^{66} - 689 q^{67} + 2522 q^{68} - 528 q^{69} + 1175 q^{70} + 341 q^{71} + 5534 q^{72} - 2260 q^{73} + 4027 q^{74} + 459 q^{75} - 1855 q^{76} + 1578 q^{77} + 5491 q^{78} + 396 q^{79} + 6183 q^{80} - 298 q^{81} + 4936 q^{82} + 1065 q^{83} + 4126 q^{84} + 144 q^{85} + 2915 q^{86} + 1564 q^{87} + 1068 q^{88} - 1984 q^{89} + 8091 q^{90} - 2186 q^{91} + 6720 q^{92} + 2540 q^{93} + 174 q^{94} + 2804 q^{95} + 6593 q^{96} - 4946 q^{97} + 7149 q^{98} + 193 q^{99}+O(q^{100}) \) 22 * q - 4 * q^2 - 13 * q^3 + 50 * q^4 - 37 * q^5 - 42 * q^6 - 52 * q^7 - 69 * q^8 + 69 * q^9 - 93 * q^10 - 77 * q^11 - 152 * q^12 - 218 * q^13 - 111 * q^14 - 142 * q^15 - 42 * q^16 - 219 * q^17 - 291 * q^18 - 476 * q^19 - 314 * q^20 - 474 * q^21 - 390 * q^22 - 202 * q^23 - 497 * q^24 - 271 * q^25 - 220 * q^26 - 244 * q^27 - 515 * q^28 - 307 * q^29 - 303 * q^30 - 1001 * q^31 - 771 * q^32 - 984 * q^33 - 1297 * q^34 - 430 * q^35 - 616 * q^36 - 922 * q^37 + 49 * q^38 - 542 * q^39 - 1344 * q^40 - 1188 * q^41 + 58 * q^42 - 192 * q^43 - 547 * q^44 - 1569 * q^45 - 1178 * q^46 - 102 * q^47 - 1209 * q^48 - 1952 * q^49 - 471 * q^50 - 834 * q^51 - 1785 * q^52 - 580 * q^53 - 1272 * q^54 - 1730 * q^55 - 804 * q^56 - 806 * q^57 - 1156 * q^58 - 1528 * q^59 + 489 * q^60 - 1631 * q^61 + 2206 * q^62 - 318 * q^63 + 327 * q^64 + 44 * q^65 + 3267 * q^66 - 689 * q^67 + 2522 * q^68 - 528 * q^69 + 1175 * q^70 + 341 * q^71 + 5534 * q^72 - 2260 * q^73 + 4027 * q^74 + 459 * q^75 - 1855 * q^76 + 1578 * q^77 + 5491 * q^78 + 396 * q^79 + 6183 * q^80 - 298 * q^81 + 4936 * q^82 + 1065 * q^83 + 4126 * q^84 + 144 * q^85 + 2915 * q^86 + 1564 * q^87 + 1068 * q^88 - 1984 * q^89 + 8091 * q^90 - 2186 * q^91 + 6720 * q^92 + 2540 * q^93 + 174 * q^94 + 2804 * q^95 + 6593 * q^96 - 4946 * q^97 + 7149 * q^98 + 193 * q^99

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 $

$2$	−4.43192	−1.56692	−0.783460	−	0.621442i	$-0.786547\pi$
$2$	−4.43192	−1.56692	−0.783460	+	0.621442i	$0.786547\pi$
$3$	6.11724	1.17726	0.588632	−	0.808401i	$-0.299667\pi$
$3$	6.11724	1.17726	0.588632	+	0.808401i	$0.299667\pi$
$4$	11.6419	1.45524
$5$	−11.7413	−1.05017	−0.525086	−	0.851049i	$-0.675967\pi$
$5$	−11.7413	−1.05017	−0.525086	+	0.851049i	$0.675967\pi$
$6$	−27.1111	−1.84468
$7$	6.42321	0.346821	0.173410	−	0.984850i	$-0.444521\pi$
$7$	6.42321	0.346821	0.173410	+	0.984850i	$0.444521\pi$
$8$	−16.1407	−0.713325
$9$	10.4206	0.385949
$10$	52.0364	1.64553
$11$	33.6179	0.921472	0.460736	−	0.887537i	$-0.347585\pi$
$11$	33.6179	0.921472	0.460736	+	0.887537i	$0.347585\pi$
$12$	71.2164	1.71320
$13$	−66.2278	−1.41295	−0.706473	−	0.707740i	$-0.749715\pi$
$13$	−66.2278	−1.41295	−0.706473	+	0.707740i	$0.749715\pi$
$14$	−28.4672	−0.543440
$15$	−71.8242	−1.23633
$16$	−21.6011	−0.337517
$17$	−29.6611	−0.423170	−0.211585	−	0.977360i	$-0.567862\pi$
$17$	−29.6611	−0.423170	−0.211585	+	0.977360i	$0.567862\pi$
$18$	−46.1834	−0.604752
$19$	−27.1748	−0.328123	−0.164061	−	0.986450i	$-0.552460\pi$
$19$	−27.1748	−0.328123	−0.164061	+	0.986450i	$0.552460\pi$
$20$	−136.691	−1.52825
$21$	39.2923	0.408299
$22$	−148.992	−1.44387
$23$	186.640	1.69205	0.846024	−	0.533145i	$-0.178990\pi$
$23$	186.640	1.69205	0.846024	+	0.533145i	$0.178990\pi$
$24$	−98.7365	−0.839771
$25$	12.8574	0.102860
$26$	293.516	2.21397
$27$	−101.420	−0.722899
$28$	74.7785	0.504707
$29$	21.8413	0.139856	0.0699281	−	0.997552i	$-0.477723\pi$
$29$	21.8413	0.139856	0.0699281	+	0.997552i	$0.477723\pi$
$30$	318.319	1.93723
$31$	−193.238	−1.11956	−0.559782	−	0.828640i	$-0.689115\pi$
$31$	−193.238	−1.11956	−0.559782	+	0.828640i	$0.689115\pi$
$32$	224.860	1.24219
$33$	205.649	1.08482
$34$	131.456	0.663073
$35$	−75.4166	−0.364221
$36$	121.316	0.561649
$37$	−67.8794	−0.301603	−0.150801	−	0.988564i	$-0.548185\pi$
$37$	−67.8794	−0.301603	−0.150801	+	0.988564i	$0.548185\pi$
$38$	120.437	0.514142
$39$	−405.131	−1.66341
$40$	189.512	0.749113
$41$	−74.4545	−0.283606	−0.141803	−	0.989895i	$-0.545290\pi$
$41$	−74.4545	−0.283606	−0.141803	+	0.989895i	$0.545290\pi$
$42$	−174.140	−0.639773
$43$	−324.175	−1.14968	−0.574840	−	0.818266i	$-0.694936\pi$
$43$	−324.175	−1.14968	−0.574840	+	0.818266i	$0.694936\pi$
$44$	391.377	1.34096
$45$	−122.351	−0.405313
$46$	−827.173	−2.65130
$47$	−202.273	−0.627757	−0.313878	−	0.949463i	$-0.601628\pi$
$47$	−202.273	−0.627757	−0.313878	+	0.949463i	$0.601628\pi$
$48$	−132.139	−0.397346
$49$	−301.742	−0.879715
$50$	−56.9832	−0.161173
$51$	−181.444	−0.498182
$52$	−771.019	−2.05617
$53$	−254.801	−0.660371	−0.330185	−	0.943916i	$-0.607111\pi$
$53$	−254.801	−0.660371	−0.330185	+	0.943916i	$0.607111\pi$
$54$	449.485	1.13273
$55$	−394.717	−0.967703
$56$	−103.675	−0.247396
$57$	−166.235	−0.386287
$58$	−96.7990	−0.219144
$59$	−474.685	−1.04744	−0.523718	−	0.851891i	$-0.675456\pi$
$59$	−474.685	−1.04744	−0.523718	+	0.851891i	$0.675456\pi$
$60$	−836.171	−1.79915
$61$	179.092	0.375909	0.187954	−	0.982178i	$-0.439814\pi$
$61$	179.092	0.375909	0.187954	+	0.982178i	$0.439814\pi$
$62$	856.414	1.75427
$63$	66.9339	0.133855
$64$	−823.752	−1.60889
$65$	777.599	1.48383
$66$	−911.420	−1.69982
$67$	294.098	0.536265	0.268133	−	0.963382i	$-0.413593\pi$
$67$	294.098	0.536265	0.268133	+	0.963382i	$0.413593\pi$
$68$	−345.313	−0.615813
$69$	1141.72	1.99199
$70$	334.241	0.570705
$71$	−126.860	−0.212050	−0.106025	−	0.994363i	$-0.533812\pi$
$71$	−126.860	−0.212050	−0.106025	+	0.994363i	$0.533812\pi$
$72$	−168.196	−0.275307
$73$	−920.278	−1.47549	−0.737743	−	0.675082i	$-0.764108\pi$
$73$	−920.278	−1.47549	−0.737743	+	0.675082i	$0.764108\pi$
$74$	300.836	0.472588
$75$	78.6521	0.121093
$76$	−316.367	−0.477497
$77$	215.935	0.319586
$78$	1795.51	2.60643
$79$	31.6274	0.0450425	0.0225213	−	0.999746i	$-0.492831\pi$
$79$	31.6274	0.0450425	0.0225213	+	0.999746i	$0.492831\pi$
$80$	253.624	0.354450
$81$	−901.767	−1.23699
$82$	329.977	0.444388
$83$	−1354.73	−1.79158	−0.895792	−	0.444474i	$-0.853390\pi$
$83$	−1354.73	−1.79158	−0.895792	+	0.444474i	$0.853390\pi$
$84$	457.438	0.594173
$85$	348.259	0.444401
$86$	1436.72	1.80146
$87$	133.609	0.164648
$88$	−542.617	−0.657309
$89$	−669.715	−0.797637	−0.398818	−	0.917030i	$-0.630580\pi$
$89$	−669.715	−0.797637	−0.398818	+	0.917030i	$0.630580\pi$
$90$	542.252	0.635093
$91$	−425.395	−0.490039
$92$	2172.85	2.46234
$93$	−1182.08	−1.31802
$94$	896.459	0.983645
$95$	319.067	0.344585
$96$	1375.52	1.46238
$97$	1611.14	1.68646	0.843228	−	0.537556i	$-0.180652\pi$
$97$	1611.14	1.68646	0.843228	+	0.537556i	$0.180652\pi$
$98$	1337.30	1.37844
$99$	350.320	0.355641
$100$	149.685	0.149685
$101$	948.093	0.934047	0.467024	−	0.884245i	$-0.345326\pi$
$101$	948.093	0.934047	0.467024	+	0.884245i	$0.345326\pi$
$102$	804.147	0.780612
$103$	−213.763	−0.204492	−0.102246	−	0.994759i	$-0.532603\pi$
$103$	−213.763	−0.204492	−0.102246	+	0.994759i	$0.532603\pi$
$104$	1068.96	1.00789
$105$	−461.342	−0.428784
$106$	1129.26	1.03475
$107$	1318.38	1.19115	0.595573	−	0.803301i	$-0.296925\pi$
$107$	1318.38	1.19115	0.595573	+	0.803301i	$0.296925\pi$
$108$	−1180.72	−1.05199
$109$	437.408	0.384368	0.192184	−	0.981359i	$-0.438443\pi$
$109$	437.408	0.384368	0.192184	+	0.981359i	$0.438443\pi$
$110$	1749.36	1.51631
$111$	−415.235	−0.355066
$112$	−138.748	−0.117058
$113$	−340.699	−0.283631	−0.141815	−	0.989893i	$-0.545294\pi$
$113$	−340.699	−0.283631	−0.141815	+	0.989893i	$0.545294\pi$
$114$	736.740	0.605281
$115$	−2191.39	−1.77694
$116$	254.275	0.203524
$117$	−690.135	−0.545325
$118$	2103.77	1.64125
$119$	−190.520	−0.146764
$120$	1159.29	0.881904
$121$	−200.834	−0.150889
$122$	−793.723	−0.589019
$123$	−455.456	−0.333879
$124$	−2249.66	−1.62924
$125$	1316.70	0.942151
$126$	−296.646	−0.209740
$127$	−194.974	−0.136230	−0.0681149	−	0.997677i	$-0.521698\pi$
$127$	−194.974	−0.136230	−0.0681149	+	0.997677i	$0.521698\pi$
$128$	1851.93	1.27882
$129$	−1983.06	−1.35348
$130$	−3446.25	−2.32505
$131$	−701.617	−0.467943	−0.233972	−	0.972243i	$-0.575172\pi$
$131$	−701.617	−0.467943	−0.233972	+	0.972243i	$0.575172\pi$
$132$	2394.15	1.57867
$133$	−174.550	−0.113800
$134$	−1303.42	−0.840285
$135$	1190.80	0.759168
$136$	478.751	0.301857
$137$	1352.62	0.843517	0.421758	−	0.906708i	$-0.361413\pi$
$137$	1352.62	0.843517	0.421758	+	0.906708i	$0.361413\pi$
$138$	−5060.02	−3.12128
$139$	2646.49	1.61491	0.807455	−	0.589930i	$-0.200845\pi$
$139$	2646.49	1.61491	0.807455	+	0.589930i	$0.200845\pi$
$140$	−877.994	−0.530029
$141$	−1237.35	−0.739035
$142$	562.234	0.332265
$143$	−2226.44	−1.30199
$144$	−225.097	−0.130264
$145$	−256.445	−0.146873
$146$	4078.60	2.31197
$147$	−1845.83	−1.03566
$148$	−790.246	−0.438904
$149$	1276.10	0.701623	0.350811	−	0.936446i	$-0.385906\pi$
$149$	1276.10	0.701623	0.350811	+	0.936446i	$0.385906\pi$
$150$	−348.580	−0.189743
$151$	717.895	0.386897	0.193448	−	0.981110i	$-0.438033\pi$
$151$	717.895	0.386897	0.193448	+	0.981110i	$0.438033\pi$
$152$	438.621	0.234058
$153$	−309.088	−0.163322
$154$	−957.007	−0.500765
$155$	2268.86	1.17573
$156$	−4716.51	−2.42066
$157$	3461.88	1.75980	0.879899	−	0.475160i	$-0.157610\pi$
$157$	3461.88	1.75980	0.879899	+	0.475160i	$0.157610\pi$
$158$	−140.170	−0.0705781
$159$	−1558.68	−0.777431
$160$	−2640.14	−1.30451
$161$	1198.83	0.586837
$162$	3996.56	1.93827
$163$	−1698.07	−0.815969	−0.407984	−	0.912989i	$-0.633768\pi$
$163$	−1698.07	−0.815969	−0.407984	+	0.912989i	$0.633768\pi$
$164$	−866.794	−0.412715
$165$	−2414.58	−1.13924
$166$	6004.07	2.80727
$167$	−3459.39	−1.60297	−0.801484	−	0.598016i	$-0.795956\pi$
$167$	−3459.39	−1.60297	−0.801484	+	0.598016i	$0.795956\pi$
$168$	−634.205	−0.291250
$169$	2189.12	0.996414
$170$	−1543.46	−0.696340
$171$	−283.179	−0.126639
$172$	−3774.02	−1.67306
$173$	2126.41	0.934499	0.467249	−	0.884126i	$-0.345245\pi$
$173$	2126.41	0.934499	0.467249	+	0.884126i	$0.345245\pi$
$174$	−592.143	−0.257990
$175$	82.5860	0.0356738
$176$	−726.184	−0.311012
$177$	−2903.76	−1.23311
$178$	2968.12	1.24983
$179$	−4035.27	−1.68498	−0.842488	−	0.538716i	$-0.818910\pi$
$179$	−4035.27	−1.68498	−0.842488	+	0.538716i	$0.818910\pi$
$180$	−1424.41	−0.589827
$181$	−465.822	−0.191294	−0.0956471	−	0.995415i	$-0.530492\pi$
$181$	−465.822	−0.191294	−0.0956471	+	0.995415i	$0.530492\pi$
$182$	1885.32	0.767852
$183$	1095.55	0.442544
$184$	−3012.50	−1.20698
$185$	796.990	0.316735
$186$	5238.89	2.06524
$187$	−997.147	−0.389939
$188$	−2354.85	−0.913537
$189$	−651.442	−0.250716
$190$	−1414.08	−0.539938
$191$	3254.72	1.23300	0.616501	−	0.787354i	$-0.288550\pi$
$191$	3254.72	1.23300	0.616501	+	0.787354i	$0.288550\pi$
$192$	−5039.09	−1.89409
$193$	−0.201305	−7.50791e−5 0	−3.75395e−5	−	1.00000i	$-0.500012\pi$
$193$	−0.201305	−7.50791e−5 0	−3.75395e−5		1.00000i	$0.500012\pi$
$194$	−7140.44	−2.64254
$195$	4756.76	1.74686
$196$	−3512.86	−1.28020
$197$	950.173	0.343640	0.171820	−	0.985128i	$-0.445035\pi$
$197$	950.173	0.343640	0.171820	+	0.985128i	$0.445035\pi$
$198$	−1552.59	−0.557262
$199$	850.338	0.302909	0.151454	−	0.988464i	$-0.451604\pi$
$199$	850.338	0.302909	0.151454	+	0.988464i	$0.451604\pi$
$200$	−207.528	−0.0733723
$201$	1799.07	0.631325
$202$	−4201.87	−1.46358
$203$	140.291	0.0485050
$204$	−2112.36	−0.724975
$205$	874.191	0.297835
$206$	947.380	0.320423
$207$	1944.91	0.653045
$208$	1430.59	0.476893
$209$	−913.562	−0.302356
$210$	2044.63	0.671871
$211$	958.773	0.312818	0.156409	−	0.987692i	$-0.450008\pi$
$211$	958.773	0.312818	0.156409	+	0.987692i	$0.450008\pi$
$212$	−2966.38	−0.960998
$213$	−776.034	−0.249638
$214$	−5842.95	−1.86643
$215$	3806.23	1.20736
$216$	1636.99	0.515662
$217$	−1241.21	−0.388288
$218$	−1938.56	−0.602274
$219$	−5629.56	−1.73703
$220$	−4595.27	−1.40824
$221$	1964.39	0.597915
$222$	1840.29	0.556360
$223$	1755.18	0.527064	0.263532	−	0.964651i	$-0.415113\pi$
$223$	1755.18	0.527064	0.263532	+	0.964651i	$0.415113\pi$
$224$	1444.32	0.430816
$225$	133.983	0.0396986
$226$	1509.95	0.444426
$227$	−250.459	−0.0732315	−0.0366158	−	0.999329i	$-0.511658\pi$
$227$	−250.459	−0.0732315	−0.0366158	+	0.999329i	$0.511658\pi$
$228$	−1935.29	−0.562140
$229$	−6835.37	−1.97246	−0.986231	−	0.165372i	$-0.947118\pi$
$229$	−6835.37	−1.97246	−0.986231	+	0.165372i	$0.947118\pi$
$230$	9712.06	2.78432
$231$	1320.93	0.376236
$232$	−352.534	−0.0997629
$233$	4731.90	1.33046	0.665229	−	0.746639i	$-0.268334\pi$
$233$	4731.90	1.33046	0.665229	+	0.746639i	$0.268334\pi$
$234$	3058.63	0.854481
$235$	2374.94	0.659252
$236$	−5526.25	−1.52427
$237$	193.472	0.0530269
$238$	844.368	0.229967
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	1551.48	0.417281
$241$	−4235.67	−1.13213	−0.566066	−	0.824360i	$-0.691535\pi$
$241$	−4235.67	−1.13213	−0.566066	+	0.824360i	$0.691535\pi$
$242$	890.079	0.236432
$243$	−2777.99	−0.733367
$244$	2084.98	0.547037
$245$	3542.84	0.923852
$246$	2018.55	0.523162
$247$	1799.73	0.463620
$248$	3118.99	0.798613
$249$	−8287.23	−2.10917
$250$	−5835.49	−1.47628
$251$	2447.32	0.615433	0.307717	−	0.951478i	$-0.400435\pi$
$251$	2447.32	0.615433	0.307717	+	0.951478i	$0.400435\pi$
$252$	779.239	0.194791
$253$	6274.45	1.55917
$254$	864.111	0.213461
$255$	2130.39	0.523177
$256$	−1617.57	−0.394915
$257$	2775.41	0.673640	0.336820	−	0.941569i	$-0.390649\pi$
$257$	2775.41	0.673640	0.336820	+	0.941569i	$0.390649\pi$
$258$	8788.75	2.12079
$259$	−436.004	−0.104602
$260$	9052.74	2.15933
$261$	227.600	0.0539774
$262$	3109.51	0.733230
$263$	391.224	0.0917259	0.0458629	−	0.998948i	$-0.485396\pi$
$263$	391.224	0.0917259	0.0458629	+	0.998948i	$0.485396\pi$
$264$	−3319.32	−0.773826
$265$	2991.69	0.693502
$266$	773.590	0.178315
$267$	−4096.81	−0.939028
$268$	3423.86	0.780395
$269$	−1160.76	−0.263097	−0.131548	−	0.991310i	$-0.541995\pi$
$269$	−1160.76	−0.263097	−0.131548	+	0.991310i	$0.541995\pi$
$270$	−5277.53	−1.18956
$271$	−1446.49	−0.324235	−0.162118	−	0.986771i	$-0.551832\pi$
$271$	−1446.49	−0.324235	−0.162118	+	0.986771i	$0.551832\pi$
$272$	640.712	0.142827
$273$	−2602.24	−0.576905
$274$	−5994.69	−1.32172
$275$	432.241	0.0947822
$276$	13291.8	2.89882
$277$	2108.86	0.457435	0.228717	−	0.973493i	$-0.426547\pi$
$277$	2108.86	0.457435	0.228717	+	0.973493i	$0.426547\pi$
$278$	−11729.0	−2.53043
$279$	−2013.66	−0.432095
$280$	1217.28	0.259808
$281$	−4475.54	−0.950137	−0.475068	−	0.879949i	$-0.657577\pi$
$281$	−4475.54	−0.950137	−0.475068	+	0.879949i	$0.657577\pi$
$282$	5483.85	1.15801
$283$	1447.46	0.304036	0.152018	−	0.988378i	$-0.451423\pi$
$283$	1447.46	0.304036	0.152018	+	0.988378i	$0.451423\pi$
$284$	−1476.90	−0.308583
$285$	1951.81	0.405668
$286$	9867.42	2.04011
$287$	−478.237	−0.0983604
$288$	2343.18	0.479421
$289$	−4033.22	−0.820927
$290$	1136.54	0.230138
$291$	9855.72	1.98540
$292$	−10713.8	−2.14718
$293$	4253.93	0.848182	0.424091	−	0.905619i	$-0.360594\pi$
$293$	4253.93	0.848182	0.424091	+	0.905619i	$0.360594\pi$
$294$	8180.58	1.62279
$295$	5573.41	1.09999
$296$	1095.62	0.215141
$297$	−3409.53	−0.666132
$298$	−5655.55	−1.09939
$299$	−12360.7	−2.39077
$300$	915.661	0.176219
$301$	−2082.24	−0.398733
$302$	−3181.65	−0.606237
$303$	5799.71	1.09962
$304$	587.005	0.110747
$305$	−2102.77	−0.394769
$306$	1369.85	0.255913
$307$	−6324.14	−1.17569	−0.587847	−	0.808972i	$-0.700024\pi$
$307$	−6324.14	−1.17569	−0.587847	+	0.808972i	$0.700024\pi$
$308$	2513.90	0.465074
$309$	−1307.64	−0.240741
$310$	−10055.4	−1.84228
$311$	−9730.54	−1.77417	−0.887087	−	0.461602i	$-0.847275\pi$
$311$	−9730.54	−1.77417	−0.887087	+	0.461602i	$0.847275\pi$
$312$	6539.10	1.18655
$313$	−5888.48	−1.06338	−0.531688	−	0.846941i	$-0.678442\pi$
$313$	−5888.48	−1.06338	−0.531688	+	0.846941i	$0.678442\pi$
$314$	−15342.8	−2.75746
$315$	−785.889	−0.140571
$316$	368.204	0.0655477
$317$	10954.9	1.94097	0.970484	−	0.241167i	$-0.0775302\pi$
$317$	10954.9	1.94097	0.970484	+	0.241167i	$0.0775302\pi$
$318$	6907.95	1.21817
$319$	734.260	0.128874
$320$	9671.90	1.68961
$321$	8064.84	1.40229
$322$	−5313.11	−0.919527
$323$	806.037	0.138852
$324$	−10498.3	−1.80012
$325$	−851.520	−0.145335
$326$	7525.70	1.27856
$327$	2675.73	0.452502
$328$	1201.75	0.202303
$329$	−1299.24	−0.217719
$330$	10701.2	1.78510
$331$	−9221.81	−1.53135	−0.765675	−	0.643228i	$-0.777595\pi$
$331$	−9221.81	−1.53135	−0.765675	+	0.643228i	$0.777595\pi$
$332$	−15771.7	−2.60718
$333$	−707.346	−0.116403
$334$	15331.7	2.51172
$335$	−3453.08	−0.563170
$336$	−848.756	−0.137808
$337$	5536.24	0.894891	0.447445	−	0.894311i	$-0.352334\pi$
$337$	5536.24	0.894891	0.447445	+	0.894311i	$0.352334\pi$
$338$	−9702.01	−1.56130
$339$	−2084.14	−0.333908
$340$	4054.41	0.646709
$341$	−6496.25	−1.03165
$342$	1255.03	0.198433
$343$	−4141.31	−0.651924
$344$	5232.41	0.820095
$345$	−13405.3	−2.09193
$346$	−9424.10	−1.46429
$347$	9164.04	1.41773	0.708864	−	0.705345i	$-0.249208\pi$
$347$	9164.04	1.41773	0.708864	+	0.705345i	$0.249208\pi$
$348$	1555.46	0.239602
$349$	−6045.01	−0.927169	−0.463585	−	0.886053i	$-0.653437\pi$
$349$	−6045.01	−0.927169	−0.463585	+	0.886053i	$0.653437\pi$
$350$	−366.015	−0.0558980
$351$	6716.82	1.02142
$352$	7559.32	1.14464
$353$	−4601.08	−0.693742	−0.346871	−	0.937913i	$-0.612756\pi$
$353$	−4601.08	−0.693742	−0.346871	+	0.937913i	$0.612756\pi$
$354$	12869.3	1.93218
$355$	1489.50	0.222689
$356$	−7796.77	−1.16075
$357$	−1165.45	−0.172780
$358$	17884.0	2.64022
$359$	11718.8	1.72283	0.861415	−	0.507901i	$-0.169579\pi$
$359$	11718.8	1.72283	0.861415	+	0.507901i	$0.169579\pi$
$360$	1974.84	0.289120
$361$	−6120.53	−0.892335
$362$	2064.48	0.299743
$363$	−1228.55	−0.177637
$364$	−4952.41	−0.713124
$365$	10805.2	1.54951
$366$	−4855.40	−0.693431
$367$	2549.66	0.362646	0.181323	−	0.983424i	$-0.441962\pi$
$367$	2549.66	0.362646	0.181323	+	0.983424i	$0.441962\pi$
$368$	−4031.62	−0.571094
$369$	−775.863	−0.109458
$370$	−3532.20	−0.496298
$371$	−1636.64	−0.229030
$372$	−13761.7	−1.91804
$373$	1826.08	0.253488	0.126744	−	0.991935i	$-0.459547\pi$
$373$	1826.08	0.253488	0.126744	+	0.991935i	$0.459547\pi$
$374$	4419.27	0.611003
$375$	8054.55	1.10916
$376$	3264.83	0.447795
$377$	−1446.50	−0.197609
$378$	2887.14	0.392853
$379$	−3976.75	−0.538976	−0.269488	−	0.963004i	$-0.586855\pi$
$379$	−3976.75	−0.538976	−0.269488	+	0.963004i	$0.586855\pi$
$380$	3714.55	0.501454
$381$	−1192.71	−0.160378
$382$	−14424.7	−1.93202
$383$	−1688.51	−0.225271	−0.112635	−	0.993636i	$-0.535929\pi$
$383$	−1688.51	−0.225271	−0.112635	+	0.993636i	$0.535929\pi$
$384$	11328.7	1.50551
$385$	−2535.35	−0.335620
$386$	0.892168	0.000117643 0
$387$	−3378.11	−0.443718
$388$	18756.7	2.45420
$389$	−9595.89	−1.25072	−0.625361	−	0.780335i	$-0.715048\pi$
$389$	−9595.89	−1.25072	−0.625361	+	0.780335i	$0.715048\pi$
$390$	−21081.6	−2.73720
$391$	−5535.95	−0.716023
$392$	4870.33	0.627523
$393$	−4291.96	−0.550892
$394$	−4211.09	−0.538456
$395$	−371.346	−0.0473024
$396$	4078.40	0.517544
$397$	−2153.46	−0.272240	−0.136120	−	0.990692i	$-0.543463\pi$
$397$	−2153.46	−0.272240	−0.136120	+	0.990692i	$0.543463\pi$
$398$	−3768.63	−0.474634
$399$	−1067.76	−0.133972
$400$	−277.735	−0.0347168
$401$	−1104.96	−0.137604	−0.0688020	−	0.997630i	$-0.521918\pi$
$401$	−1104.96	−0.137604	−0.0688020	+	0.997630i	$0.521918\pi$
$402$	−7973.32	−0.989237
$403$	12797.7	1.58188
$404$	11037.6	1.35926
$405$	10587.9	1.29905
$406$	−621.760	−0.0760035
$407$	−2281.97	−0.277919
$408$	2928.64	0.355366
$409$	−4624.39	−0.559074	−0.279537	−	0.960135i	$-0.590181\pi$
$409$	−4624.39	−0.559074	−0.279537	+	0.960135i	$0.590181\pi$
$410$	−3874.34	−0.466683
$411$	8274.28	0.993041
$412$	−2488.61	−0.297585
$413$	−3049.00	−0.363273
$414$	−8619.67	−1.02327
$415$	15906.3	1.88147
$416$	−14892.0	−1.75514
$417$	16189.2	1.90117
$418$	4048.83	0.473768
$419$	15049.7	1.75472	0.877361	−	0.479832i	$-0.159302\pi$
$419$	15049.7	1.75472	0.877361	+	0.479832i	$0.159302\pi$
$420$	−5370.90	−0.623984
$421$	3597.85	0.416505	0.208252	−	0.978075i	$-0.433222\pi$
$421$	3597.85	0.416505	0.208252	+	0.978075i	$0.433222\pi$
$422$	−4249.21	−0.490161
$423$	−2107.81	−0.242282
$424$	4112.67	0.471059
$425$	−381.366	−0.0435270
$426$	3439.32	0.391164
$427$	1150.35	0.130373
$428$	15348.5	1.73340
$429$	−13619.7	−1.53278
$430$	−16868.9	−1.89184
$431$	8421.15	0.941143	0.470571	−	0.882362i	$-0.344048\pi$
$431$	8421.15	0.941143	0.470571	+	0.882362i	$0.344048\pi$
$432$	2190.78	0.243991
$433$	−456.823	−0.0507010	−0.0253505	−	0.999679i	$-0.508070\pi$
$433$	−456.823	−0.0507010	−0.0253505	+	0.999679i	$0.508070\pi$
$434$	5500.93	0.608417
$435$	−1568.73	−0.172908
$436$	5092.27	0.559348
$437$	−5071.91	−0.555200
$438$	24949.8	2.72180
$439$	−3913.92	−0.425515	−0.212758	−	0.977105i	$-0.568244\pi$
$439$	−3913.92	−0.425515	−0.212758	+	0.977105i	$0.568244\pi$
$440$	6371.01	0.690287
$441$	−3144.35	−0.339526
$442$	−8706.03	−0.936886
$443$	9716.83	1.04212	0.521062	−	0.853519i	$-0.325536\pi$
$443$	9716.83	1.04212	0.521062	+	0.853519i	$0.325536\pi$
$444$	−4834.13	−0.516706
$445$	7863.31	0.837655
$446$	−7778.80	−0.825868
$447$	7806.18	0.825995
$448$	−5291.13	−0.557997
$449$	−7757.41	−0.815356	−0.407678	−	0.913126i	$-0.633661\pi$
$449$	−7757.41	−0.815356	−0.407678	+	0.913126i	$0.633661\pi$
$450$	−593.801	−0.0622045
$451$	−2503.01	−0.261335
$452$	−3966.39	−0.412750
$453$	4391.53	0.455480
$454$	1110.01	0.114748
$455$	4994.68	0.514624
$456$	2683.15	0.275548
$457$	2081.07	0.213017	0.106508	−	0.994312i	$-0.466033\pi$
$457$	2081.07	0.213017	0.106508	+	0.994312i	$0.466033\pi$
$458$	30293.8	3.09069
$459$	3008.23	0.305909
$460$	−25512.0	−2.58587
$461$	15231.8	1.53886	0.769432	−	0.638728i	$-0.220539\pi$
$461$	15231.8	1.53886	0.769432	+	0.638728i	$0.220539\pi$
$462$	−5854.24	−0.589533
$463$	6789.34	0.681485	0.340742	−	0.940157i	$-0.389322\pi$
$463$	6789.34	0.681485	0.340742	+	0.940157i	$0.389322\pi$
$464$	−471.796	−0.0472038
$465$	13879.1	1.38415
$466$	−20971.4	−2.08472
$467$	9501.47	0.941489	0.470745	−	0.882269i	$-0.343985\pi$
$467$	9501.47	0.941489	0.470745	+	0.882269i	$0.343985\pi$
$468$	−8034.50	−0.793579
$469$	1889.05	0.185988
$470$	−10525.6	−1.03300
$471$	21177.2	2.07175
$472$	7661.75	0.747163
$473$	−10898.1	−1.05940
$474$	−857.454	−0.0830890
$475$	−349.399	−0.0337506
$476$	−2218.01	−0.213577
$477$	−2655.19	−0.254870
$478$	−1059.23	−0.101356
$479$	−13193.9	−1.25855	−0.629273	−	0.777184i	$-0.716647\pi$
$479$	−13193.9	−1.25855	−0.629273	+	0.777184i	$0.716647\pi$
$480$	−16150.4	−1.53575
$481$	4495.50	0.426148
$482$	18772.2	1.77396
$483$	7333.51	0.690862
$484$	−2338.09	−0.219580
$485$	−18916.8	−1.77107
$486$	12311.8	1.14913
$487$	1521.50	0.141572	0.0707862	−	0.997492i	$-0.477449\pi$
$487$	1521.50	0.141572	0.0707862	+	0.997492i	$0.477449\pi$
$488$	−2890.68	−0.268145
$489$	−10387.5	−0.960610
$490$	−15701.6	−1.44760
$491$	10578.2	0.972279	0.486139	−	0.873881i	$-0.338405\pi$
$491$	10578.2	0.972279	0.486139	+	0.873881i	$0.338405\pi$
$492$	−5302.38	−0.485874
$493$	−647.838	−0.0591829
$494$	−7976.26	−0.726455
$495$	−4113.20	−0.373484
$496$	4174.14	0.377872
$497$	−814.849	−0.0735432
$498$	36728.4	3.30489
$499$	−12634.3	−1.13344	−0.566722	−	0.823909i	$-0.691789\pi$
$499$	−12634.3	−1.13344	−0.566722	+	0.823909i	$0.691789\pi$
$500$	15328.9	1.37106
$501$	−21161.9	−1.88712
$502$	−10846.3	−0.964335
$503$	−12052.3	−1.06836	−0.534182	−	0.845369i	$-0.679380\pi$
$503$	−12052.3	−1.06836	−0.534182	+	0.845369i	$0.679380\pi$
$504$	−1080.36	−0.0954822
$505$	−11131.8	−0.980910
$506$	−27807.9	−2.44310
$507$	13391.4	1.17304
$508$	−2269.88	−0.198247
$509$	−13175.9	−1.14737	−0.573684	−	0.819076i	$-0.694486\pi$
$509$	−13175.9	−1.14737	−0.573684	+	0.819076i	$0.694486\pi$
$510$	−9441.71	−0.819776
$511$	−5911.14	−0.511729
$512$	−7646.46	−0.660018
$513$	2756.07	0.237200
$514$	−12300.4	−1.05554
$515$	2509.85	0.214752
$516$	−23086.6	−1.96963
$517$	−6800.01	−0.578460
$518$	1932.33	0.163903
$519$	13007.8	1.10015
$520$	−12551.0	−1.05846
$521$	−16729.8	−1.40681	−0.703404	−	0.710790i	$-0.748338\pi$
$521$	−16729.8	−1.40681	−0.703404	+	0.710790i	$0.748338\pi$
$522$	−1008.71	−0.0845783
$523$	14265.5	1.19271	0.596353	−	0.802722i	$-0.296616\pi$
$523$	14265.5	1.19271	0.596353	+	0.802722i	$0.296616\pi$
$524$	−8168.17	−0.680970
$525$	505.199	0.0419975
$526$	−1733.87	−0.143727
$527$	5731.65	0.473766
$528$	−4442.24	−0.366143
$529$	22667.4	1.86303
$530$	−13258.9	−1.08666
$531$	−4946.52	−0.404258
$532$	−2032.09	−0.165606
$533$	4930.96	0.400720
$534$	18156.7	1.47138
$535$	−15479.5	−1.25091
$536$	−4746.94	−0.382531
$537$	−24684.7	−1.98366
$538$	5144.41	0.412251
$539$	−10144.0	−0.810633
$540$	13863.2	1.10477
$541$	−11927.0	−0.947843	−0.473922	−	0.880567i	$-0.657162\pi$
$541$	−11927.0	−0.947843	−0.473922	+	0.880567i	$0.657162\pi$
$542$	6410.71	0.508051
$543$	−2849.54	−0.225204
$544$	−6669.60	−0.525656
$545$	−5135.73	−0.403652
$546$	11532.9	0.903964
$547$	−21208.1	−1.65776	−0.828880	−	0.559427i	$-0.811021\pi$
$547$	−21208.1	−1.65776	−0.828880	+	0.559427i	$0.811021\pi$
$548$	15747.0	1.22752
$549$	1866.26	0.145082
$550$	−1915.66	−0.148516
$551$	−593.534	−0.0458900
$552$	−18428.2	−1.42093
$553$	203.149	0.0156217
$554$	−9346.32	−0.716764
$555$	4875.38	0.372880
$556$	30810.2	2.35008
$557$	9468.58	0.720281	0.360140	−	0.932898i	$-0.382729\pi$
$557$	9468.58	0.720281	0.360140	+	0.932898i	$0.382729\pi$
$558$	8924.38	0.677059
$559$	21469.4	1.62444
$560$	1629.08	0.122931
$561$	−6099.79	−0.459061
$562$	19835.2	1.48879
$563$	8933.50	0.668743	0.334372	−	0.942441i	$-0.391476\pi$
$563$	8933.50	0.668743	0.334372	+	0.942441i	$0.391476\pi$
$564$	−14405.2	−1.07547
$565$	4000.24	0.297861
$566$	−6415.01	−0.476401
$567$	−5792.24	−0.429015
$568$	2047.61	0.151260
$569$	−3255.65	−0.239866	−0.119933	−	0.992782i	$-0.538268\pi$
$569$	−3255.65	−0.239866	−0.119933	+	0.992782i	$0.538268\pi$
$570$	−8650.27	−0.635649
$571$	23018.3	1.68701	0.843507	−	0.537118i	$-0.180487\pi$
$571$	23018.3	1.68701	0.843507	+	0.537118i	$0.180487\pi$
$572$	−25920.1	−1.89471
$573$	19909.9	1.45157
$574$	2119.51	0.154123
$575$	2399.71	0.174043
$576$	−8584.02	−0.620950
$577$	−14141.7	−1.02033	−0.510163	−	0.860078i	$-0.670415\pi$
$577$	−14141.7	−1.02033	−0.510163	+	0.860078i	$0.670415\pi$
$578$	17874.9	1.28633
$579$	−1.23143	−8.83878e−5 0
$580$	−2985.51	−0.213735
$581$	−8701.74	−0.621358
$582$	−43679.8	−3.11097
$583$	−8565.90	−0.608513
$584$	14853.9	1.05250
$585$	8103.07	0.572685
$586$	−18853.1	−1.32903
$587$	−22822.8	−1.60477	−0.802384	−	0.596808i	$-0.796436\pi$
$587$	−22822.8	−1.60477	−0.802384	+	0.596808i	$0.796436\pi$
$588$	−21489.0	−1.50713
$589$	5251.20	0.367355
$590$	−24700.9	−1.72359
$591$	5812.43	0.404554
$592$	1466.27	0.101796
$593$	−20090.6	−1.39127	−0.695633	−	0.718398i	$-0.744876\pi$
$593$	−20090.6	−1.39127	−0.695633	+	0.718398i	$0.744876\pi$
$594$	15110.8	1.04378
$595$	2236.94	0.154127
$596$	14856.2	1.02103
$597$	5201.72	0.356603
$598$	54781.9	3.74615
$599$	−23493.3	−1.60252	−0.801259	−	0.598318i	$-0.795836\pi$
$599$	−23493.3	−1.60252	−0.801259	+	0.598318i	$0.795836\pi$
$600$	−1269.50	−0.0863785
$601$	26271.2	1.78307	0.891536	−	0.452951i	$-0.149629\pi$
$601$	26271.2	1.78307	0.891536	+	0.452951i	$0.149629\pi$
$602$	9228.34	0.624783
$603$	3064.69	0.206971
$604$	8357.67	0.563028
$605$	2358.04	0.158460
$606$	−25703.9	−1.72302
$607$	14888.7	0.995575	0.497787	−	0.867299i	$-0.334146\pi$
$607$	14888.7	0.995575	0.497787	+	0.867299i	$0.334146\pi$
$608$	−6110.53	−0.407590
$609$	858.196	0.0571032
$610$	9319.32	0.618571
$611$	13396.1	0.886986
$612$	−3598.38	−0.237673
$613$	8028.48	0.528984	0.264492	−	0.964388i	$-0.414796\pi$
$613$	8028.48	0.528984	0.264492	+	0.964388i	$0.414796\pi$
$614$	28028.1	1.84222
$615$	5347.63	0.350630
$616$	−3485.34	−0.227968
$617$	14781.3	0.964463	0.482232	−	0.876044i	$-0.339826\pi$
$617$	14781.3	0.964463	0.482232	+	0.876044i	$0.339826\pi$
$618$	5795.35	0.377222
$619$	−6891.50	−0.447484	−0.223742	−	0.974648i	$-0.571827\pi$
$619$	−6891.50	−0.447484	−0.223742	+	0.974648i	$0.571827\pi$
$620$	26413.8	1.71098
$621$	−18929.0	−1.22318
$622$	43125.0	2.77999
$623$	−4301.72	−0.276637
$624$	8751.27	0.561428
$625$	−17066.9	−1.09228
$626$	26097.3	1.66622
$627$	−5588.48	−0.355953
$628$	40303.0	2.56093
$629$	2013.38	0.127629
$630$	3483.00	0.220263
$631$	−13868.2	−0.874936	−0.437468	−	0.899234i	$-0.644125\pi$
$631$	−13868.2	−0.874936	−0.437468	+	0.899234i	$0.644125\pi$
$632$	−510.488	−0.0321300
$633$	5865.05	0.368270
$634$	−48551.1	−3.04134
$635$	2289.25	0.143065
$636$	−18146.0	−1.13135
$637$	19983.7	1.24299
$638$	−3254.18	−0.201935
$639$	−1321.96	−0.0818405
$640$	−21744.0	−1.34298
$641$	22387.7	1.37951	0.689753	−	0.724045i	$-0.257719\pi$
$641$	22387.7	1.37951	0.689753	+	0.724045i	$0.257719\pi$
$642$	−35742.7	−2.19728
$643$	24966.6	1.53124	0.765619	−	0.643294i	$-0.222433\pi$
$643$	24966.6	1.53124	0.765619	+	0.643294i	$0.222433\pi$
$644$	13956.6	0.853989
$645$	23283.6	1.42138
$646$	−3572.29	−0.217569
$647$	−4254.36	−0.258510	−0.129255	−	0.991611i	$-0.541259\pi$
$647$	−4254.36	−0.258510	−0.129255	+	0.991611i	$0.541259\pi$
$648$	14555.2	0.882377
$649$	−15957.9	−0.965184
$650$	3773.87	0.227728
$651$	−7592.76	−0.457118
$652$	−19768.8	−1.18743
$653$	−4413.54	−0.264495	−0.132247	−	0.991217i	$-0.542219\pi$
$653$	−4413.54	−0.264495	−0.132247	+	0.991217i	$0.542219\pi$
$654$	−11858.6	−0.709035
$655$	8237.87	0.491420
$656$	1608.30	0.0957217
$657$	−9589.88	−0.569462
$658$	5758.14	0.341149
$659$	−18203.8	−1.07606	−0.538028	−	0.842927i	$-0.680830\pi$
$659$	−18203.8	−1.07606	−0.538028	+	0.842927i	$0.680830\pi$
$660$	−28110.4	−1.65787
$661$	−10835.4	−0.637589	−0.318795	−	0.947824i	$-0.603278\pi$
$661$	−10835.4	−0.637589	−0.318795	+	0.947824i	$0.603278\pi$
$662$	40870.3	2.39950
$663$	12016.7	0.703904
$664$	21866.4	1.27798
$665$	2049.43	0.119509
$666$	3134.90	0.182395
$667$	4076.46	0.236643
$668$	−40274.0	−2.33270
$669$	10736.8	0.620493
$670$	15303.8	0.882443
$671$	6020.72	0.346389
$672$	8835.26	0.507184
$673$	9652.28	0.552850	0.276425	−	0.961036i	$-0.410850\pi$
$673$	9652.28	0.552850	0.276425	+	0.961036i	$0.410850\pi$
$674$	−24536.2	−1.40222
$675$	−1304.00	−0.0743571
$676$	25485.6	1.45002
$677$	−28447.0	−1.61493	−0.807466	−	0.589914i	$-0.799161\pi$
$677$	−28447.0	−1.61493	−0.807466	+	0.589914i	$0.799161\pi$
$678$	9236.73	0.523207
$679$	10348.7	0.584898
$680$	−5621.15	−0.317002
$681$	−1532.12	−0.0862128
$682$	28790.9	1.61651
$683$	−7578.79	−0.424589	−0.212295	−	0.977206i	$-0.568094\pi$
$683$	−7578.79	−0.424589	−0.212295	+	0.977206i	$0.568094\pi$
$684$	−3296.75	−0.184290
$685$	−15881.4	−0.885837
$686$	18354.0	1.02151
$687$	−41813.6	−2.32211
$688$	7002.53	0.388036
$689$	16874.9	0.933068
$690$	59411.0	3.27788
$691$	−24228.3	−1.33385	−0.666924	−	0.745126i	$-0.732389\pi$
$691$	−24228.3	−1.33385	−0.666924	+	0.745126i	$0.732389\pi$
$692$	24755.5	1.35992
$693$	2250.18	0.123344
$694$	−40614.3	−2.22147
$695$	−31073.2	−1.69593
$696$	−2156.54	−0.117447
$697$	2208.41	0.120013
$698$	26791.0	1.45280
$699$	28946.1	1.56630
$700$	961.460	0.0519140
$701$	−19186.0	−1.03373	−0.516866	−	0.856067i	$-0.672901\pi$
$701$	−19186.0	−1.03373	−0.516866	+	0.856067i	$0.672901\pi$
$702$	−29768.4	−1.60048
$703$	1844.61	0.0989628
$704$	−27692.9	−1.48255
$705$	14528.1	0.776114
$706$	20391.6	1.08704
$707$	6089.80	0.323947
$708$	−33805.4	−1.79447
$709$	−3588.22	−0.190069	−0.0950343	−	0.995474i	$-0.530296\pi$
$709$	−3588.22	−0.190069	−0.0950343	+	0.995474i	$0.530296\pi$
$710$	−6601.34	−0.348935
$711$	329.578	0.0173841
$712$	10809.7	0.568974
$713$	−36065.9	−1.89436
$714$	5165.20	0.270732
$715$	26141.3	1.36731
$716$	−46978.3	−2.45204
$717$	1462.02	0.0761508
$718$	−51936.9	−2.69954
$719$	−33269.0	−1.72562	−0.862811	−	0.505526i	$-0.831298\pi$
$719$	−33269.0	−1.72562	−0.862811	+	0.505526i	$0.831298\pi$
$720$	2642.92	0.136800
$721$	−1373.04	−0.0709221
$722$	27125.7	1.39822
$723$	−25910.6	−1.33282
$724$	−5423.06	−0.278379
$725$	280.824	0.0143856
$726$	5444.83	0.278342
$727$	37966.0	1.93684	0.968419	−	0.249329i	$-0.0802101\pi$
$727$	37966.0	1.93684	0.968419	+	0.249329i	$0.0802101\pi$
$728$	6866.17	0.349557
$729$	7354.10	0.373627
$730$	−47887.9	−2.42796
$731$	9615.41	0.486510
$732$	12754.3	0.644007
$733$	9538.53	0.480646	0.240323	−	0.970693i	$-0.422747\pi$
$733$	9538.53	0.480646	0.240323	+	0.970693i	$0.422747\pi$
$734$	−11299.9	−0.568238
$735$	21672.4	1.08762
$736$	41967.8	2.10184
$737$	9886.97	0.494153
$738$	3438.56	0.171511
$739$	5437.53	0.270667	0.135333	−	0.990800i	$-0.456789\pi$
$739$	5437.53	0.270667	0.135333	+	0.990800i	$0.456789\pi$
$740$	9278.50	0.460925
$741$	11009.4	0.545802
$742$	7253.47	0.358872
$743$	16270.6	0.803377	0.401688	−	0.915776i	$-0.368423\pi$
$743$	16270.6	0.803377	0.401688	+	0.915776i	$0.368423\pi$
$744$	19079.6	0.940178
$745$	−14983.0	−0.736824
$746$	−8093.04	−0.397195
$747$	−14117.2	−0.691460
$748$	−11608.7	−0.567455
$749$	8468.23	0.413114
$750$	−35697.1	−1.73797
$751$	14167.4	0.688384	0.344192	−	0.938899i	$-0.388153\pi$
$751$	14167.4	0.688384	0.344192	+	0.938899i	$0.388153\pi$
$752$	4369.32	0.211878
$753$	14970.9	0.724527
$754$	6410.78	0.309638
$755$	−8429.00	−0.406308
$756$	−7584.03	−0.364853
$757$	−13061.8	−0.627134	−0.313567	−	0.949566i	$-0.601524\pi$
$757$	−13061.8	−0.627134	−0.313567	+	0.949566i	$0.601524\pi$
$758$	17624.7	0.844533
$759$	38382.3	1.83556
$760$	−5149.96	−0.245801
$761$	23975.5	1.14207	0.571034	−	0.820927i	$-0.306543\pi$
$761$	23975.5	1.14207	0.571034	+	0.820927i	$0.306543\pi$
$762$	5285.98	0.251300
$763$	2809.56	0.133307
$764$	37891.2	1.79431
$765$	3629.08	0.171516
$766$	7483.34	0.352982
$767$	31437.4	1.47997
$768$	−9895.07	−0.464919
$769$	25602.3	1.20058	0.600289	−	0.799783i	$-0.295052\pi$
$769$	25602.3	1.20058	0.600289	+	0.799783i	$0.295052\pi$
$770$	11236.5	0.525889
$771$	16977.9	0.793052
$772$	−2.34358	−0.000109258 0
$773$	5621.51	0.261568	0.130784	−	0.991411i	$-0.458251\pi$
$773$	5621.51	0.261568	0.130784	+	0.991411i	$0.458251\pi$
$774$	14971.5	0.695271
$775$	−2484.54	−0.115158
$776$	−26004.9	−1.20299
$777$	−2667.14	−0.123144
$778$	42528.2	1.95978
$779$	2023.29	0.0930576
$780$	55377.8	2.54211
$781$	−4264.78	−0.195398
$782$	24534.9	1.12195
$783$	−2215.15	−0.101102
$784$	6517.96	0.296919
$785$	−40646.9	−1.84809
$786$	19021.6	0.863205
$787$	26335.0	1.19281	0.596405	−	0.802684i	$-0.296595\pi$
$787$	26335.0	1.19281	0.596405	+	0.802684i	$0.296595\pi$
$788$	11061.8	0.500078
$789$	2393.21	0.107986
$790$	1645.78	0.0741191
$791$	−2188.38	−0.0983689
$792$	−5654.41	−0.253688
$793$	−11860.9	−0.531138
$794$	9543.97	0.426578
$795$	18300.9	0.816435
$796$	9899.56	0.440805
$797$	8800.99	0.391150	0.195575	−	0.980689i	$-0.437343\pi$
$797$	8800.99	0.391150	0.195575	+	0.980689i	$0.437343\pi$
$798$	4732.24	0.209924
$799$	5999.65	0.265648
$800$	2891.12	0.127771
$801$	−6978.85	−0.307847
$802$	4897.11	0.215615
$803$	−30937.9	−1.35962
$804$	20944.6	0.918730
$805$	−14075.8	−0.616280
$806$	−56718.4	−2.47869
$807$	−7100.67	−0.309734
$808$	−15302.9	−0.666279
$809$	28919.9	1.25682	0.628412	−	0.777881i	$-0.283705\pi$
$809$	28919.9	1.25682	0.628412	+	0.777881i	$0.283705\pi$
$810$	−46924.7	−2.03551
$811$	−2510.47	−0.108699	−0.0543493	−	0.998522i	$-0.517308\pi$
$811$	−2510.47	−0.108699	−0.0543493	+	0.998522i	$0.517308\pi$
$812$	1633.26	0.0705865
$813$	−8848.51	−0.381711
$814$	10113.5	0.435476
$815$	19937.5	0.856907
$816$	3919.39	0.168145
$817$	8809.41	0.377236
$818$	20494.9	0.876024
$819$	−4432.88	−0.189130
$820$	10177.3	0.433421
$821$	−25922.6	−1.10196	−0.550978	−	0.834520i	$-0.685745\pi$
$821$	−25922.6	−1.10196	−0.550978	+	0.834520i	$0.685745\pi$
$822$	−36670.9	−1.55602
$823$	−26762.9	−1.13353	−0.566765	−	0.823879i	$-0.691805\pi$
$823$	−26762.9	−1.13353	−0.566765	+	0.823879i	$0.691805\pi$
$824$	3450.28	0.145869
$825$	2644.12	0.111584
$826$	13512.9	0.569220
$827$	−45365.0	−1.90749	−0.953745	−	0.300616i	$-0.902808\pi$
$827$	−45365.0	−1.90749	−0.953745	+	0.300616i	$0.902808\pi$
$828$	22642.4	0.950337
$829$	−35434.7	−1.48456	−0.742279	−	0.670091i	$-0.766255\pi$
$829$	−35434.7	−1.48456	−0.742279	+	0.670091i	$0.766255\pi$
$830$	−70495.5	−2.94811
$831$	12900.4	0.538521
$832$	54555.3	2.27327
$833$	8950.02	0.372269
$834$	−71749.3	−2.97899
$835$	40617.7	1.68339
$836$	−10635.6	−0.440001
$837$	19598.2	0.809333
$838$	−66699.3	−2.74951
$839$	12190.2	0.501610	0.250805	−	0.968038i	$-0.419305\pi$
$839$	12190.2	0.501610	0.250805	+	0.968038i	$0.419305\pi$
$840$	7446.38	0.305862
$841$	−23912.0	−0.980440
$842$	−15945.4	−0.652630
$843$	−27378.0	−1.11856
$844$	11162.0	0.455226
$845$	−25703.1	−1.04641
$846$	9341.67	0.379637
$847$	−1290.00	−0.0523316
$848$	5503.98	0.222886
$849$	8854.43	0.357931
$850$	1690.19	0.0682034
$851$	−12669.0	−0.510326
$852$	−9034.53	−0.363284
$853$	−9104.99	−0.365473	−0.182737	−	0.983162i	$-0.558496\pi$
$853$	−9104.99	−0.365473	−0.182737	+	0.983162i	$0.558496\pi$
$854$	−5098.25	−0.204284
$855$	3324.88	0.132992
$856$	−21279.6	−0.849673
$857$	−30408.4	−1.21206	−0.606028	−	0.795443i	$-0.707238\pi$
$857$	−30408.4	−1.21206	−0.606028	+	0.795443i	$0.707238\pi$
$858$	60361.4	2.40175
$859$	−18161.1	−0.721361	−0.360680	−	0.932689i	$-0.617455\pi$
$859$	−18161.1	−0.721361	−0.360680	+	0.932689i	$0.617455\pi$
$860$	44311.8	1.75700
$861$	−2925.49	−0.115796
$862$	−37321.9	−1.47470
$863$	20392.3	0.804358	0.402179	−	0.915561i	$-0.368253\pi$
$863$	20392.3	0.804358	0.402179	+	0.915561i	$0.368253\pi$
$864$	−22805.3	−0.897976
$865$	−24966.8	−0.981384
$866$	2024.61	0.0794444
$867$	−24672.2	−0.966448
$868$	−14450.0	−0.565053
$869$	1063.25	0.0415054
$870$	6952.51	0.270933
$871$	−19477.5	−0.757713
$872$	−7060.07	−0.274179
$873$	16789.1	0.650887
$874$	22478.3	0.869954
$875$	8457.41	0.326757
$876$	−65538.9	−2.52780
$877$	41135.1	1.58385	0.791923	−	0.610620i	$-0.209080\pi$
$877$	41135.1	1.58385	0.791923	+	0.610620i	$0.209080\pi$
$878$	17346.2	0.666748
$879$	26022.3	0.998534
$880$	8526.32	0.326616
$881$	−5084.72	−0.194448	−0.0972238	−	0.995263i	$-0.530996\pi$
$881$	−5084.72	−0.194448	−0.0972238	+	0.995263i	$0.530996\pi$
$882$	13935.5	0.532010
$883$	−40168.5	−1.53089	−0.765446	−	0.643500i	$-0.777482\pi$
$883$	−40168.5	−1.53089	−0.765446	+	0.643500i	$0.777482\pi$
$884$	22869.3	0.870110
$885$	34093.9	1.29498
$886$	−43064.2	−1.63292
$887$	21339.6	0.807795	0.403898	−	0.914804i	$-0.367655\pi$
$887$	21339.6	0.807795	0.403898	+	0.914804i	$0.367655\pi$
$888$	6702.18	0.253277
$889$	−1252.36	−0.0472473
$890$	−34849.5	−1.31254
$891$	−30315.6	−1.13985
$892$	20433.6	0.767005
$893$	5496.74	0.205981
$894$	−34596.4	−1.29427
$895$	47379.3	1.76951
$896$	11895.3	0.443520
$897$	−75613.7	−2.81457
$898$	34380.2	1.27760
$899$	−4220.57	−0.156578
$900$	1559.82	0.0577709
$901$	7557.70	0.279449
$902$	11093.1	0.409491
$903$	−12737.6	−0.469414
$904$	5499.12	0.202321
$905$	5469.34	0.200892
$906$	−19462.9	−0.713700
$907$	−50489.5	−1.84838	−0.924188	−	0.381938i	$-0.875257\pi$
$907$	−50489.5	−1.84838	−0.924188	+	0.381938i	$0.875257\pi$
$908$	−2915.82	−0.106569
$909$	9879.73	0.360495
$910$	−22136.0	−0.806376
$911$	40812.6	1.48428	0.742142	−	0.670243i	$-0.233810\pi$
$911$	40812.6	1.48428	0.742142	+	0.670243i	$0.233810\pi$
$912$	3590.85	0.130378
$913$	−45543.4	−1.65089
$914$	−9223.16	−0.333780
$915$	−12863.2	−0.464747
$916$	−79576.8	−2.87041
$917$	−4506.63	−0.162292
$918$	−13332.2	−0.479335
$919$	29107.2	1.04478	0.522392	−	0.852705i	$-0.325040\pi$
$919$	29107.2	1.04478	0.522392	+	0.852705i	$0.325040\pi$
$920$	35370.5	1.26754
$921$	−38686.3	−1.38410
$922$	−67506.2	−2.41128
$923$	8401.67	0.299615
$924$	15378.1	0.547514
$925$	−872.756	−0.0310227
$926$	−30089.8	−1.06783
$927$	−2227.54	−0.0789235
$928$	4911.23	0.173728
$929$	11312.5	0.399516	0.199758	−	0.979845i	$-0.435984\pi$
$929$	11312.5	0.399516	0.199758	+	0.979845i	$0.435984\pi$
$930$	−61511.2	−2.16885
$931$	8199.80	0.288655
$932$	55088.4	1.93614
$933$	−59524.0	−2.08867
$934$	−42109.8	−1.47524
$935$	11707.8	0.409503
$936$	11139.3	0.388994
$937$	−40082.3	−1.39747	−0.698736	−	0.715379i	$-0.746254\pi$
$937$	−40082.3	−1.39747	−0.698736	+	0.715379i	$0.746254\pi$
$938$	−8372.13	−0.291428
$939$	−36021.2	−1.25187
$940$	27648.9	0.959370
$941$	−2708.22	−0.0938209	−0.0469105	−	0.998899i	$-0.514938\pi$
$941$	−2708.22	−0.0938209	−0.0469105	+	0.998899i	$0.514938\pi$
$942$	−93855.6	−3.24626
$943$	−13896.2	−0.479875
$944$	10253.7	0.353527
$945$	7648.75	0.263295
$946$	48299.5	1.65999
$947$	−15460.5	−0.530515	−0.265257	−	0.964178i	$-0.585457\pi$
$947$	−15460.5	−0.530515	−0.265257	+	0.964178i	$0.585457\pi$
$948$	2252.39	0.0771669
$949$	60948.0	2.08478
$950$	1548.51	0.0528845
$951$	67013.6	2.28503
$952$	3075.12	0.104690
$953$	−12109.6	−0.411614	−0.205807	−	0.978593i	$-0.565982\pi$
$953$	−12109.6	−0.411614	−0.205807	+	0.978593i	$0.565982\pi$
$954$	11767.6	0.399361
$955$	−38214.6	−1.29486
$956$	2782.42	0.0941316
$957$	4491.65	0.151718
$958$	58474.2	1.97204
$959$	8688.14	0.292549
$960$	59165.3	1.98912
$961$	7549.81	0.253426
$962$	−19923.7	−0.667740
$963$	13738.3	0.459722
$964$	−49311.3	−1.64752
$965$	2.36358	7.88459e−5 0
$966$	−32501.5	−1.08253
$967$	−39971.5	−1.32926	−0.664632	−	0.747171i	$-0.731411\pi$
$967$	−39971.5	−1.32926	−0.664632	+	0.747171i	$0.731411\pi$
$968$	3241.60	0.107633
$969$	4930.72	0.163465
$970$	83837.8	2.77512
$971$	9447.30	0.312233	0.156117	−	0.987739i	$-0.450102\pi$
$971$	9447.30	0.312233	0.156117	+	0.987739i	$0.450102\pi$
$972$	−32341.1	−1.06722
$973$	16999.0	0.560084
$974$	−6743.17	−0.221833
$975$	−5208.95	−0.171097
$976$	−3868.59	−0.126875
$977$	36175.0	1.18459	0.592293	−	0.805722i	$-0.298223\pi$
$977$	36175.0	1.18459	0.592293	+	0.805722i	$0.298223\pi$
$978$	46036.5	1.50520
$979$	−22514.4	−0.735000
$980$	41245.4	1.34443
$981$	4558.07	0.148347
$982$	−46881.9	−1.52348
$983$	−36082.6	−1.17076	−0.585380	−	0.810759i	$-0.699055\pi$
$983$	−36082.6	−1.17076	−0.585380	+	0.810759i	$0.699055\pi$
$984$	7351.38	0.238164
$985$	−11156.2	−0.360880
$986$	2871.17	0.0927349
$987$	−7947.78	−0.256313
$988$	20952.3	0.674678
$989$	−60504.0	−1.94531
$990$	18229.4	0.585220
$991$	−54791.5	−1.75632	−0.878158	−	0.478371i	$-0.841227\pi$
$991$	−54791.5	−1.75632	−0.878158	+	0.478371i	$0.841227\pi$
$992$	−43451.4	−1.39071
$993$	−56412.1	−1.80280
$994$	3611.35	0.115236
$995$	−9984.05	−0.318106
$996$	−96479.3	−3.06934
$997$	3815.64	0.121206	0.0606030	−	0.998162i	$-0.480698\pi$
$997$	3815.64	0.121206	0.0606030	+	0.998162i	$0.480698\pi$
$998$	55994.2	1.77602
$999$	6884.33	0.218029

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	239.4.a.a.1.3	✓	22
3.2	odd	2		2151.4.a.a.1.20		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.a.1.3	✓	22	1.1	even	1	trivial
2151.4.a.a.1.20		22	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	239.a (trivial)

\(f(q)\)	\(=\)	\(q-4.43192 q^{2} +6.11724 q^{3} +11.6419 q^{4} -11.7413 q^{5} -27.1111 q^{6} +6.42321 q^{7} -16.1407 q^{8} +10.4206 q^{9} +O(q^{10})\)	\(q-4.43192 q^{2} +6.11724 q^{3} +11.6419 q^{4} -11.7413 q^{5} -27.1111 q^{6} +6.42321 q^{7} -16.1407 q^{8} +10.4206 q^{9} +52.0364 q^{10} +33.6179 q^{11} +71.2164 q^{12} -66.2278 q^{13} -28.4672 q^{14} -71.8242 q^{15} -21.6011 q^{16} -29.6611 q^{17} -46.1834 q^{18} -27.1748 q^{19} -136.691 q^{20} +39.2923 q^{21} -148.992 q^{22} +186.640 q^{23} -98.7365 q^{24} +12.8574 q^{25} +293.516 q^{26} -101.420 q^{27} +74.7785 q^{28} +21.8413 q^{29} +318.319 q^{30} -193.238 q^{31} +224.860 q^{32} +205.649 q^{33} +131.456 q^{34} -75.4166 q^{35} +121.316 q^{36} -67.8794 q^{37} +120.437 q^{38} -405.131 q^{39} +189.512 q^{40} -74.4545 q^{41} -174.140 q^{42} -324.175 q^{43} +391.377 q^{44} -122.351 q^{45} -827.173 q^{46} -202.273 q^{47} -132.139 q^{48} -301.742 q^{49} -56.9832 q^{50} -181.444 q^{51} -771.019 q^{52} -254.801 q^{53} +449.485 q^{54} -394.717 q^{55} -103.675 q^{56} -166.235 q^{57} -96.7990 q^{58} -474.685 q^{59} -836.171 q^{60} +179.092 q^{61} +856.414 q^{62} +66.9339 q^{63} -823.752 q^{64} +777.599 q^{65} -911.420 q^{66} +294.098 q^{67} -345.313 q^{68} +1141.72 q^{69} +334.241 q^{70} -126.860 q^{71} -168.196 q^{72} -920.278 q^{73} +300.836 q^{74} +78.6521 q^{75} -316.367 q^{76} +215.935 q^{77} +1795.51 q^{78} +31.6274 q^{79} +253.624 q^{80} -901.767 q^{81} +329.977 q^{82} -1354.73 q^{83} +457.438 q^{84} +348.259 q^{85} +1436.72 q^{86} +133.609 q^{87} -542.617 q^{88} -669.715 q^{89} +542.252 q^{90} -425.395 q^{91} +2172.85 q^{92} -1182.08 q^{93} +896.459 q^{94} +319.067 q^{95} +1375.52 q^{96} +1611.14 q^{97} +1337.30 q^{98} +350.320 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 4 q^{2} - 13 q^{3} + 50 q^{4} - 37 q^{5} - 42 q^{6} - 52 q^{7} - 69 q^{8} + 69 q^{9}+O(q^{10}) \) 22 * q - 4 * q^2 - 13 * q^3 + 50 * q^4 - 37 * q^5 - 42 * q^6 - 52 * q^7 - 69 * q^8 + 69 * q^9	\( 22 q - 4 q^{2} - 13 q^{3} + 50 q^{4} - 37 q^{5} - 42 q^{6} - 52 q^{7} - 69 q^{8} + 69 q^{9} - 93 q^{10} - 77 q^{11} - 152 q^{12} - 218 q^{13} - 111 q^{14} - 142 q^{15} - 42 q^{16} - 219 q^{17} - 291 q^{18} - 476 q^{19} - 314 q^{20} - 474 q^{21} - 390 q^{22} - 202 q^{23} - 497 q^{24} - 271 q^{25} - 220 q^{26} - 244 q^{27} - 515 q^{28} - 307 q^{29} - 303 q^{30} - 1001 q^{31} - 771 q^{32} - 984 q^{33} - 1297 q^{34} - 430 q^{35} - 616 q^{36} - 922 q^{37} + 49 q^{38} - 542 q^{39} - 1344 q^{40} - 1188 q^{41} + 58 q^{42} - 192 q^{43} - 547 q^{44} - 1569 q^{45} - 1178 q^{46} - 102 q^{47} - 1209 q^{48} - 1952 q^{49} - 471 q^{50} - 834 q^{51} - 1785 q^{52} - 580 q^{53} - 1272 q^{54} - 1730 q^{55} - 804 q^{56} - 806 q^{57} - 1156 q^{58} - 1528 q^{59} + 489 q^{60} - 1631 q^{61} + 2206 q^{62} - 318 q^{63} + 327 q^{64} + 44 q^{65} + 3267 q^{66} - 689 q^{67} + 2522 q^{68} - 528 q^{69} + 1175 q^{70} + 341 q^{71} + 5534 q^{72} - 2260 q^{73} + 4027 q^{74} + 459 q^{75} - 1855 q^{76} + 1578 q^{77} + 5491 q^{78} + 396 q^{79} + 6183 q^{80} - 298 q^{81} + 4936 q^{82} + 1065 q^{83} + 4126 q^{84} + 144 q^{85} + 2915 q^{86} + 1564 q^{87} + 1068 q^{88} - 1984 q^{89} + 8091 q^{90} - 2186 q^{91} + 6720 q^{92} + 2540 q^{93} + 174 q^{94} + 2804 q^{95} + 6593 q^{96} - 4946 q^{97} + 7149 q^{98} + 193 q^{99}+O(q^{100}) \) 22 * q - 4 * q^2 - 13 * q^3 + 50 * q^4 - 37 * q^5 - 42 * q^6 - 52 * q^7 - 69 * q^8 + 69 * q^9 - 93 * q^10 - 77 * q^11 - 152 * q^12 - 218 * q^13 - 111 * q^14 - 142 * q^15 - 42 * q^16 - 219 * q^17 - 291 * q^18 - 476 * q^19 - 314 * q^20 - 474 * q^21 - 390 * q^22 - 202 * q^23 - 497 * q^24 - 271 * q^25 - 220 * q^26 - 244 * q^27 - 515 * q^28 - 307 * q^29 - 303 * q^30 - 1001 * q^31 - 771 * q^32 - 984 * q^33 - 1297 * q^34 - 430 * q^35 - 616 * q^36 - 922 * q^37 + 49 * q^38 - 542 * q^39 - 1344 * q^40 - 1188 * q^41 + 58 * q^42 - 192 * q^43 - 547 * q^44 - 1569 * q^45 - 1178 * q^46 - 102 * q^47 - 1209 * q^48 - 1952 * q^49 - 471 * q^50 - 834 * q^51 - 1785 * q^52 - 580 * q^53 - 1272 * q^54 - 1730 * q^55 - 804 * q^56 - 806 * q^57 - 1156 * q^58 - 1528 * q^59 + 489 * q^60 - 1631 * q^61 + 2206 * q^62 - 318 * q^63 + 327 * q^64 + 44 * q^65 + 3267 * q^66 - 689 * q^67 + 2522 * q^68 - 528 * q^69 + 1175 * q^70 + 341 * q^71 + 5534 * q^72 - 2260 * q^73 + 4027 * q^74 + 459 * q^75 - 1855 * q^76 + 1578 * q^77 + 5491 * q^78 + 396 * q^79 + 6183 * q^80 - 298 * q^81 + 4936 * q^82 + 1065 * q^83 + 4126 * q^84 + 144 * q^85 + 2915 * q^86 + 1564 * q^87 + 1068 * q^88 - 1984 * q^89 + 8091 * q^90 - 2186 * q^91 + 6720 * q^92 + 2540 * q^93 + 174 * q^94 + 2804 * q^95 + 6593 * q^96 - 4946 * q^97 + 7149 * q^98 + 193 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more