LMFDB - Embedded newform 2151.4.a.a.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.32829 q^{2} -2.57908 q^{4} +14.1221 q^{5} -19.8005 q^{7} +24.6311 q^{8} +O(q^{10})$	\(q-2.32829 q^{2} -2.57908 q^{4} +14.1221 q^{5} -19.8005 q^{7} +24.6311 q^{8} -32.8804 q^{10} +12.7034 q^{11} +41.5067 q^{13} +46.1012 q^{14} -36.7157 q^{16} +32.9417 q^{17} -96.7273 q^{19} -36.4221 q^{20} -29.5772 q^{22} -32.1384 q^{23} +74.4346 q^{25} -96.6396 q^{26} +51.0670 q^{28} +43.8488 q^{29} +187.507 q^{31} -111.564 q^{32} -76.6977 q^{34} -279.625 q^{35} -411.234 q^{37} +225.209 q^{38} +347.844 q^{40} -152.148 q^{41} +327.866 q^{43} -32.7631 q^{44} +74.8273 q^{46} +17.3791 q^{47} +49.0586 q^{49} -173.305 q^{50} -107.049 q^{52} -448.191 q^{53} +179.399 q^{55} -487.708 q^{56} -102.093 q^{58} -92.8880 q^{59} +92.6079 q^{61} -436.571 q^{62} +553.479 q^{64} +586.164 q^{65} -191.991 q^{67} -84.9592 q^{68} +651.047 q^{70} +500.561 q^{71} -1025.52 q^{73} +957.471 q^{74} +249.468 q^{76} -251.534 q^{77} -771.727 q^{79} -518.504 q^{80} +354.244 q^{82} +1268.18 q^{83} +465.207 q^{85} -763.367 q^{86} +312.900 q^{88} +979.322 q^{89} -821.853 q^{91} +82.8874 q^{92} -40.4636 q^{94} -1366.00 q^{95} +1290.20 q^{97} -114.223 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

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 $

$2$	−2.32829	−0.823174	−0.411587	−	0.911371i	$-0.635025\pi$
$2$	−2.32829	−0.823174	−0.411587	+	0.911371i	$0.635025\pi$
$3$	0	0
$3$	0	0
$4$	−2.57908	−0.322385
$5$	14.1221	1.26312	0.631561	−	0.775326i	$-0.282415\pi$
$5$	14.1221	1.26312	0.631561	+	0.775326i	$0.282415\pi$
$6$	0	0
$7$	−19.8005	−1.06912	−0.534562	−	0.845129i	$-0.679524\pi$
$7$	−19.8005	−1.06912	−0.534562	+	0.845129i	$0.679524\pi$
$8$	24.6311	1.08855
$9$	0	0
$10$	−32.8804	−1.03977
$11$	12.7034	0.348202	0.174101	−	0.984728i	$-0.444298\pi$
$11$	12.7034	0.348202	0.174101	+	0.984728i	$0.444298\pi$
$12$	0	0
$13$	41.5067	0.885531	0.442765	−	0.896637i	$-0.353997\pi$
$13$	41.5067	0.885531	0.442765	+	0.896637i	$0.353997\pi$
$14$	46.1012	0.880076
$15$	0	0
$16$	−36.7157	−0.573683
$17$	32.9417	0.469972	0.234986	−	0.971999i	$-0.424496\pi$
$17$	32.9417	0.469972	0.234986	+	0.971999i	$0.424496\pi$
$18$	0	0
$19$	−96.7273	−1.16794	−0.583968	−	0.811777i	$-0.698500\pi$
$19$	−96.7273	−1.16794	−0.583968	+	0.811777i	$0.698500\pi$
$20$	−36.4221	−0.407212
$21$	0	0
$22$	−29.5772	−0.286631
$23$	−32.1384	−0.291361	−0.145681	−	0.989332i	$-0.546537\pi$
$23$	−32.1384	−0.291361	−0.145681	+	0.989332i	$0.546537\pi$
$24$	0	0
$25$	74.4346	0.595477
$26$	−96.6396	−0.728946
$27$	0	0
$28$	51.0670	0.344670
$29$	43.8488	0.280777	0.140388	−	0.990097i	$-0.455165\pi$
$29$	43.8488	0.280777	0.140388	+	0.990097i	$0.455165\pi$
$30$	0	0
$31$	187.507	1.08636	0.543182	−	0.839615i	$-0.317219\pi$
$31$	187.507	1.08636	0.543182	+	0.839615i	$0.317219\pi$
$32$	−111.564	−0.616312
$33$	0	0
$34$	−76.6977	−0.386869
$35$	−279.625	−1.35044
$36$	0	0
$37$	−411.234	−1.82720	−0.913601	−	0.406611i	$-0.866711\pi$
$37$	−411.234	−1.82720	−0.913601	+	0.406611i	$0.866711\pi$
$38$	225.209	0.961413
$39$	0	0
$40$	347.844	1.37497
$41$	−152.148	−0.579549	−0.289775	−	0.957095i	$-0.593580\pi$
$41$	−152.148	−0.579549	−0.289775	+	0.957095i	$0.593580\pi$
$42$	0	0
$43$	327.866	1.16277	0.581385	−	0.813628i	$-0.302511\pi$
$43$	327.866	1.16277	0.581385	+	0.813628i	$0.302511\pi$
$44$	−32.7631	−0.112255
$45$	0	0
$46$	74.8273	0.239841
$47$	17.3791	0.0539364	0.0269682	−	0.999636i	$-0.491415\pi$
$47$	17.3791	0.0539364	0.0269682	+	0.999636i	$0.491415\pi$
$48$	0	0
$49$	49.0586	0.143028
$50$	−173.305	−0.490181
$51$	0	0
$52$	−107.049	−0.285482
$53$	−448.191	−1.16158	−0.580790	−	0.814053i	$-0.697256\pi$
$53$	−448.191	−1.16158	−0.580790	+	0.814053i	$0.697256\pi$
$54$	0	0
$55$	179.399	0.439822
$56$	−487.708	−1.16380
$57$	0	0
$58$	−102.093	−0.231128
$59$	−92.8880	−0.204966	−0.102483	−	0.994735i	$-0.532679\pi$
$59$	−92.8880	−0.204966	−0.102483	+	0.994735i	$0.532679\pi$
$60$	0	0
$61$	92.6079	0.194381	0.0971904	−	0.995266i	$-0.469014\pi$
$61$	92.6079	0.194381	0.0971904	+	0.995266i	$0.469014\pi$
$62$	−436.571	−0.894267
$63$	0	0
$64$	553.479	1.08101
$65$	586.164	1.11853
$66$	0	0
$67$	−191.991	−0.350082	−0.175041	−	0.984561i	$-0.556006\pi$
$67$	−191.991	−0.350082	−0.175041	+	0.984561i	$0.556006\pi$
$68$	−84.9592	−0.151512
$69$	0	0
$70$	651.047	1.11164
$71$	500.561	0.836699	0.418350	−	0.908286i	$-0.362609\pi$
$71$	500.561	0.836699	0.418350	+	0.908286i	$0.362609\pi$
$72$	0	0
$73$	−1025.52	−1.64423	−0.822114	−	0.569323i	$-0.807205\pi$
$73$	−1025.52	−1.64423	−0.822114	+	0.569323i	$0.807205\pi$
$74$	957.471	1.50411
$75$	0	0
$76$	249.468	0.376525
$77$	−251.534	−0.372272
$78$	0	0
$79$	−771.727	−1.09906	−0.549532	−	0.835472i	$-0.685194\pi$
$79$	−771.727	−1.09906	−0.549532	+	0.835472i	$0.685194\pi$
$80$	−518.504	−0.724631
$81$	0	0
$82$	354.244	0.477070
$83$	1268.18	1.67712	0.838558	−	0.544812i	$-0.183399\pi$
$83$	1268.18	1.67712	0.838558	+	0.544812i	$0.183399\pi$
$84$	0	0
$85$	465.207	0.593632
$86$	−763.367	−0.957162
$87$	0	0
$88$	312.900	0.379036
$89$	979.322	1.16638	0.583191	−	0.812335i	$-0.301804\pi$
$89$	979.322	1.16638	0.583191	+	0.812335i	$0.301804\pi$
$90$	0	0
$91$	−821.853	−0.946743
$92$	82.8874	0.0939306
$93$	0	0
$94$	−40.4636	−0.0443990
$95$	−1366.00	−1.47524
$96$	0	0
$97$	1290.20	1.35051	0.675255	−	0.737584i	$-0.264034\pi$
$97$	1290.20	1.35051	0.675255	+	0.737584i	$0.264034\pi$
$98$	−114.223	−0.117737
$99$	0	0
$100$	−191.973	−0.191973
$101$	−1258.12	−1.23949	−0.619743	−	0.784805i	$-0.712763\pi$
$101$	−1258.12	−1.23949	−0.619743	+	0.784805i	$0.712763\pi$
$102$	0	0
$103$	96.2850	0.0921091	0.0460546	−	0.998939i	$-0.485335\pi$
$103$	96.2850	0.0921091	0.0460546	+	0.998939i	$0.485335\pi$
$104$	1022.36	0.963947
$105$	0	0
$106$	1043.52	0.956183
$107$	149.130	0.134738	0.0673688	−	0.997728i	$-0.478540\pi$
$107$	149.130	0.134738	0.0673688	+	0.997728i	$0.478540\pi$
$108$	0	0
$109$	398.668	0.350325	0.175163	−	0.984540i	$-0.443955\pi$
$109$	398.668	0.350325	0.175163	+	0.984540i	$0.443955\pi$
$110$	−417.693	−0.362050
$111$	0	0
$112$	726.988	0.613339
$113$	−639.051	−0.532008	−0.266004	−	0.963972i	$-0.585703\pi$
$113$	−639.051	−0.532008	−0.266004	+	0.963972i	$0.585703\pi$
$114$	0	0
$115$	−453.862	−0.368025
$116$	−113.090	−0.0905182
$117$	0	0
$118$	216.270	0.168722
$119$	−652.261	−0.502459
$120$	0	0
$121$	−1169.62	−0.878755
$122$	−215.618	−0.160009
$123$	0	0
$124$	−483.596	−0.350228
$125$	−714.091	−0.510962
$126$	0	0
$127$	1198.81	0.837617	0.418809	−	0.908075i	$-0.362448\pi$
$127$	1198.81	0.837617	0.418809	+	0.908075i	$0.362448\pi$
$128$	−396.144	−0.273551
$129$	0	0
$130$	−1364.76	−0.920747
$131$	−2280.78	−1.52117	−0.760583	−	0.649241i	$-0.775087\pi$
$131$	−2280.78	−1.52117	−0.760583	+	0.649241i	$0.775087\pi$
$132$	0	0
$133$	1915.25	1.24867
$134$	447.011	0.288178
$135$	0	0
$136$	811.391	0.511590
$137$	783.692	0.488725	0.244362	−	0.969684i	$-0.421421\pi$
$137$	783.692	0.488725	0.244362	+	0.969684i	$0.421421\pi$
$138$	0	0
$139$	−1926.16	−1.17536	−0.587679	−	0.809094i	$-0.699958\pi$
$139$	−1926.16	−1.17536	−0.587679	+	0.809094i	$0.699958\pi$
$140$	721.175	0.435360
$141$	0	0
$142$	−1165.45	−0.688749
$143$	527.278	0.308344
$144$	0	0
$145$	619.239	0.354655
$146$	2387.72	1.35348
$147$	0	0
$148$	1060.61	0.589063
$149$	2864.18	1.57479	0.787393	−	0.616452i	$-0.211431\pi$
$149$	2864.18	1.57479	0.787393	+	0.616452i	$0.211431\pi$
$150$	0	0
$151$	−2120.34	−1.14272	−0.571360	−	0.820700i	$-0.693584\pi$
$151$	−2120.34	−1.14272	−0.571360	+	0.820700i	$0.693584\pi$
$152$	−2382.50	−1.27136
$153$	0	0
$154$	585.643	0.306444
$155$	2648.00	1.37221
$156$	0	0
$157$	−1299.81	−0.660741	−0.330371	−	0.943851i	$-0.607174\pi$
$157$	−1299.81	−0.660741	−0.330371	+	0.943851i	$0.607174\pi$
$158$	1796.80	0.904721
$159$	0	0
$160$	−1575.53	−0.778477
$161$	636.355	0.311502
$162$	0	0
$163$	−543.969	−0.261392	−0.130696	−	0.991422i	$-0.541721\pi$
$163$	−543.969	−0.261392	−0.130696	+	0.991422i	$0.541721\pi$
$164$	392.402	0.186838
$165$	0	0
$166$	−2952.68	−1.38056
$167$	1149.40	0.532595	0.266297	−	0.963891i	$-0.414200\pi$
$167$	1149.40	0.532595	0.266297	+	0.963891i	$0.414200\pi$
$168$	0	0
$169$	−474.190	−0.215835
$170$	−1083.13	−0.488663
$171$	0	0
$172$	−845.593	−0.374860
$173$	−1040.00	−0.457049	−0.228525	−	0.973538i	$-0.573390\pi$
$173$	−1040.00	−0.457049	−0.228525	+	0.973538i	$0.573390\pi$
$174$	0	0
$175$	−1473.84	−0.636639
$176$	−466.415	−0.199758
$177$	0	0
$178$	−2280.14	−0.960135
$179$	382.041	0.159526	0.0797628	−	0.996814i	$-0.474584\pi$
$179$	382.041	0.159526	0.0797628	+	0.996814i	$0.474584\pi$
$180$	0	0
$181$	−751.930	−0.308787	−0.154394	−	0.988009i	$-0.549342\pi$
$181$	−751.930	−0.308787	−0.154394	+	0.988009i	$0.549342\pi$
$182$	1913.51	0.779334
$183$	0	0
$184$	−791.604	−0.317162
$185$	−5807.51	−2.30798
$186$	0	0
$187$	418.472	0.163645
$188$	−44.8222	−0.0173883
$189$	0	0
$190$	3180.43	1.21438
$191$	1375.93	0.521249	0.260624	−	0.965440i	$-0.416072\pi$
$191$	1375.93	0.521249	0.260624	+	0.965440i	$0.416072\pi$
$192$	0	0
$193$	2308.22	0.860876	0.430438	−	0.902620i	$-0.358359\pi$
$193$	2308.22	0.860876	0.430438	+	0.902620i	$0.358359\pi$
$194$	−3003.94	−1.11170
$195$	0	0
$196$	−126.526	−0.0461101
$197$	1842.88	0.666495	0.333248	−	0.942839i	$-0.391855\pi$
$197$	1842.88	0.666495	0.333248	+	0.942839i	$0.391855\pi$
$198$	0	0
$199$	3898.40	1.38869	0.694347	−	0.719640i	$-0.255693\pi$
$199$	3898.40	1.38869	0.694347	+	0.719640i	$0.255693\pi$
$200$	1833.41	0.648208
$201$	0	0
$202$	2929.27	1.02031
$203$	−868.227	−0.300185
$204$	0	0
$205$	−2148.65	−0.732041
$206$	−224.179	−0.0758218
$207$	0	0
$208$	−1523.95	−0.508014
$209$	−1228.77	−0.406678
$210$	0	0
$211$	1695.28	0.553118	0.276559	−	0.960997i	$-0.410806\pi$
$211$	1695.28	0.553118	0.276559	+	0.960997i	$0.410806\pi$
$212$	1155.92	0.374476
$213$	0	0
$214$	−347.217	−0.110912
$215$	4630.17	1.46872
$216$	0	0
$217$	−3712.73	−1.16146
$218$	−928.213	−0.288378
$219$	0	0
$220$	−462.685	−0.141792
$221$	1367.30	0.416175
$222$	0	0
$223$	−1644.66	−0.493877	−0.246938	−	0.969031i	$-0.579425\pi$
$223$	−1644.66	−0.493877	−0.246938	+	0.969031i	$0.579425\pi$
$224$	2209.03	0.658915
$225$	0	0
$226$	1487.89	0.437935
$227$	6564.05	1.91926	0.959628	−	0.281271i	$-0.0907559\pi$
$227$	6564.05	1.91926	0.959628	+	0.281271i	$0.0907559\pi$
$228$	0	0
$229$	3163.91	0.913000	0.456500	−	0.889723i	$-0.349103\pi$
$229$	3163.91	0.913000	0.456500	+	0.889723i	$0.349103\pi$
$230$	1056.72	0.302949
$231$	0	0
$232$	1080.05	0.305640
$233$	−3117.11	−0.876431	−0.438216	−	0.898870i	$-0.644389\pi$
$233$	−3117.11	−0.876431	−0.438216	+	0.898870i	$0.644389\pi$
$234$	0	0
$235$	245.431	0.0681282
$236$	239.566	0.0660779
$237$	0	0
$238$	1518.65	0.413611
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−670.681	−0.179263	−0.0896314	−	0.995975i	$-0.528569\pi$
$241$	−670.681	−0.179263	−0.0896314	+	0.995975i	$0.528569\pi$
$242$	2723.22	0.723368
$243$	0	0
$244$	−238.843	−0.0626655
$245$	692.813	0.180662
$246$	0	0
$247$	−4014.84	−1.03424
$248$	4618.52	1.18257
$249$	0	0
$250$	1662.61	0.420610
$251$	3166.29	0.796233	0.398116	−	0.917335i	$-0.369664\pi$
$251$	3166.29	0.796233	0.398116	+	0.917335i	$0.369664\pi$
$252$	0	0
$253$	−408.267	−0.101453
$254$	−2791.18	−0.689504
$255$	0	0
$256$	−3505.50	−0.855835
$257$	2294.89	0.557010	0.278505	−	0.960435i	$-0.410161\pi$
$257$	2294.89	0.557010	0.278505	+	0.960435i	$0.410161\pi$
$258$	0	0
$259$	8142.63	1.95351
$260$	−1511.76	−0.360598
$261$	0	0
$262$	5310.31	1.25218
$263$	−5069.34	−1.18855	−0.594276	−	0.804261i	$-0.702561\pi$
$263$	−5069.34	−1.18855	−0.594276	+	0.804261i	$0.702561\pi$
$264$	0	0
$265$	−6329.42	−1.46722
$266$	−4459.24	−1.02787
$267$	0	0
$268$	495.161	0.112861
$269$	−1453.21	−0.329383	−0.164691	−	0.986345i	$-0.552663\pi$
$269$	−1453.21	−0.329383	−0.164691	+	0.986345i	$0.552663\pi$
$270$	0	0
$271$	4468.72	1.00168	0.500840	−	0.865540i	$-0.333024\pi$
$271$	4468.72	1.00168	0.500840	+	0.865540i	$0.333024\pi$
$272$	−1209.48	−0.269615
$273$	0	0
$274$	−1824.66	−0.402305
$275$	945.574	0.207346
$276$	0	0
$277$	−3211.30	−0.696565	−0.348283	−	0.937390i	$-0.613235\pi$
$277$	−3211.30	−0.696565	−0.348283	+	0.937390i	$0.613235\pi$
$278$	4484.66	0.967524
$279$	0	0
$280$	−6887.48	−1.47002
$281$	−7361.03	−1.56271	−0.781356	−	0.624085i	$-0.785472\pi$
$281$	−7361.03	−1.56271	−0.781356	+	0.624085i	$0.785472\pi$
$282$	0	0
$283$	−7508.51	−1.57715	−0.788577	−	0.614936i	$-0.789182\pi$
$283$	−7508.51	−1.57715	−0.788577	+	0.614936i	$0.789182\pi$
$284$	−1290.99	−0.269739
$285$	0	0
$286$	−1227.65	−0.253821
$287$	3012.60	0.619611
$288$	0	0
$289$	−3827.85	−0.779126
$290$	−1441.77	−0.291943
$291$	0	0
$292$	2644.91	0.530074
$293$	−5280.69	−1.05291	−0.526453	−	0.850204i	$-0.676478\pi$
$293$	−5280.69	−1.05291	−0.526453	+	0.850204i	$0.676478\pi$
$294$	0	0
$295$	−1311.78	−0.258897
$296$	−10129.2	−1.98901
$297$	0	0
$298$	−6668.64	−1.29632
$299$	−1333.96	−0.258010
$300$	0	0
$301$	−6491.90	−1.24315
$302$	4936.76	0.940657
$303$	0	0
$304$	3551.41	0.670024
$305$	1307.82	0.245527
$306$	0	0
$307$	3224.93	0.599532	0.299766	−	0.954013i	$-0.403091\pi$
$307$	3224.93	0.599532	0.299766	+	0.954013i	$0.403091\pi$
$308$	648.726	0.120015
$309$	0	0
$310$	−6165.31	−1.12957
$311$	−8628.60	−1.57326	−0.786629	−	0.617426i	$-0.788175\pi$
$311$	−8628.60	−1.57326	−0.786629	+	0.617426i	$0.788175\pi$
$312$	0	0
$313$	−9550.57	−1.72470	−0.862349	−	0.506315i	$-0.831007\pi$
$313$	−9550.57	−1.72470	−0.862349	+	0.506315i	$0.831007\pi$
$314$	3026.34	0.543905
$315$	0	0
$316$	1990.35	0.354322
$317$	−8563.55	−1.51728	−0.758639	−	0.651512i	$-0.774135\pi$
$317$	−8563.55	−1.51728	−0.758639	+	0.651512i	$0.774135\pi$
$318$	0	0
$319$	557.030	0.0977670
$320$	7816.31	1.36545
$321$	0	0
$322$	−1481.62	−0.256420
$323$	−3186.36	−0.548897
$324$	0	0
$325$	3089.54	0.527313
$326$	1266.52	0.215171
$327$	0	0
$328$	−3747.58	−0.630870
$329$	−344.115	−0.0576647
$330$	0	0
$331$	−6669.32	−1.10749	−0.553744	−	0.832687i	$-0.686802\pi$
$331$	−6669.32	−1.10749	−0.553744	+	0.832687i	$0.686802\pi$
$332$	−3270.73	−0.540677
$333$	0	0
$334$	−2676.14	−0.438418
$335$	−2711.33	−0.442196
$336$	0	0
$337$	−7433.04	−1.20149	−0.600747	−	0.799439i	$-0.705130\pi$
$337$	−7433.04	−1.20149	−0.600747	+	0.799439i	$0.705130\pi$
$338$	1104.05	0.177670
$339$	0	0
$340$	−1199.81	−0.191378
$341$	2381.98	0.378275
$342$	0	0
$343$	5820.18	0.916210
$344$	8075.72	1.26574
$345$	0	0
$346$	2421.41	0.376231
$347$	−8690.42	−1.34446	−0.672228	−	0.740344i	$-0.734663\pi$
$347$	−8690.42	−1.34446	−0.672228	+	0.740344i	$0.734663\pi$
$348$	0	0
$349$	9707.21	1.48887	0.744434	−	0.667696i	$-0.232719\pi$
$349$	9707.21	1.48887	0.744434	+	0.667696i	$0.232719\pi$
$350$	3431.52	0.524065
$351$	0	0
$352$	−1417.25	−0.214601
$353$	−4602.09	−0.693894	−0.346947	−	0.937885i	$-0.612782\pi$
$353$	−4602.09	−0.693894	−0.346947	+	0.937885i	$0.612782\pi$
$354$	0	0
$355$	7068.99	1.05685
$356$	−2525.75	−0.376024
$357$	0	0
$358$	−889.501	−0.131317
$359$	1961.44	0.288358	0.144179	−	0.989552i	$-0.453946\pi$
$359$	1961.44	0.288358	0.144179	+	0.989552i	$0.453946\pi$
$360$	0	0
$361$	2497.17	0.364072
$362$	1750.71	0.254186
$363$	0	0
$364$	2119.63	0.305216
$365$	−14482.6	−2.07686
$366$	0	0
$367$	−7392.96	−1.05152	−0.525762	−	0.850632i	$-0.676220\pi$
$367$	−7392.96	−1.05152	−0.525762	+	0.850632i	$0.676220\pi$
$368$	1179.98	0.167149
$369$	0	0
$370$	13521.5	1.89987
$371$	8874.40	1.24188
$372$	0	0
$373$	−9453.88	−1.31234	−0.656171	−	0.754613i	$-0.727825\pi$
$373$	−9453.88	−1.31234	−0.656171	+	0.754613i	$0.727825\pi$
$374$	−974.323	−0.134709
$375$	0	0
$376$	428.068	0.0587126
$377$	1820.02	0.248636
$378$	0	0
$379$	−958.264	−0.129875	−0.0649376	−	0.997889i	$-0.520685\pi$
$379$	−958.264	−0.129875	−0.0649376	+	0.997889i	$0.520685\pi$
$380$	3523.01	0.475597
$381$	0	0
$382$	−3203.55	−0.429078
$383$	−3166.70	−0.422483	−0.211241	−	0.977434i	$-0.567751\pi$
$383$	−3166.70	−0.422483	−0.211241	+	0.977434i	$0.567751\pi$
$384$	0	0
$385$	−3552.19	−0.470225
$386$	−5374.19	−0.708650
$387$	0	0
$388$	−3327.52	−0.435384
$389$	−5239.67	−0.682935	−0.341467	−	0.939894i	$-0.610924\pi$
$389$	−5239.67	−0.682935	−0.341467	+	0.939894i	$0.610924\pi$
$390$	0	0
$391$	−1058.69	−0.136932
$392$	1208.37	0.155694
$393$	0	0
$394$	−4290.75	−0.548641
$395$	−10898.4	−1.38825
$396$	0	0
$397$	−6175.90	−0.780754	−0.390377	−	0.920655i	$-0.627655\pi$
$397$	−6175.90	−0.780754	−0.390377	+	0.920655i	$0.627655\pi$
$398$	−9076.59	−1.14314
$399$	0	0
$400$	−2732.92	−0.341615
$401$	−2171.02	−0.270363	−0.135182	−	0.990821i	$-0.543162\pi$
$401$	−2171.02	−0.270363	−0.135182	+	0.990821i	$0.543162\pi$
$402$	0	0
$403$	7782.82	0.962010
$404$	3244.80	0.399592
$405$	0	0
$406$	2021.48	0.247105
$407$	−5224.08	−0.636236
$408$	0	0
$409$	−3851.55	−0.465641	−0.232820	−	0.972520i	$-0.574795\pi$
$409$	−3851.55	−0.465641	−0.232820	+	0.972520i	$0.574795\pi$
$410$	5002.68	0.602597
$411$	0	0
$412$	−248.327	−0.0296946
$413$	1839.23	0.219134
$414$	0	0
$415$	17909.4	2.11840
$416$	−4630.68	−0.545763
$417$	0	0
$418$	2860.92	0.334766
$419$	−4973.58	−0.579893	−0.289947	−	0.957043i	$-0.593638\pi$
$419$	−4973.58	−0.579893	−0.289947	+	0.957043i	$0.593638\pi$
$420$	0	0
$421$	−4057.35	−0.469699	−0.234849	−	0.972032i	$-0.575460\pi$
$421$	−4057.35	−0.469699	−0.234849	+	0.972032i	$0.575460\pi$
$422$	−3947.10	−0.455312
$423$	0	0
$424$	−11039.5	−1.26444
$425$	2452.00	0.279858
$426$	0	0
$427$	−1833.68	−0.207817
$428$	−384.618	−0.0434374
$429$	0	0
$430$	−10780.4	−1.20901
$431$	1495.34	0.167118	0.0835589	−	0.996503i	$-0.473371\pi$
$431$	1495.34	0.167118	0.0835589	+	0.996503i	$0.473371\pi$
$432$	0	0
$433$	−12069.6	−1.33956	−0.669778	−	0.742561i	$-0.733611\pi$
$433$	−12069.6	−1.33956	−0.669778	+	0.742561i	$0.733611\pi$
$434$	8644.31	0.956083
$435$	0	0
$436$	−1028.20	−0.112940
$437$	3108.66	0.340291
$438$	0	0
$439$	15678.3	1.70452	0.852260	−	0.523119i	$-0.175232\pi$
$439$	15678.3	1.70452	0.852260	+	0.523119i	$0.175232\pi$
$440$	4418.81	0.478769
$441$	0	0
$442$	−3183.47	−0.342584
$443$	14930.5	1.60128	0.800642	−	0.599143i	$-0.204492\pi$
$443$	14930.5	1.60128	0.800642	+	0.599143i	$0.204492\pi$
$444$	0	0
$445$	13830.1	1.47328
$446$	3829.24	0.406546
$447$	0	0
$448$	−10959.2	−1.15574
$449$	−11966.3	−1.25773	−0.628867	−	0.777513i	$-0.716481\pi$
$449$	−11966.3	−1.25773	−0.628867	+	0.777513i	$0.716481\pi$
$450$	0	0
$451$	−1932.80	−0.201800
$452$	1648.16	0.171511
$453$	0	0
$454$	−15283.0	−1.57988
$455$	−11606.3	−1.19585
$456$	0	0
$457$	−8113.60	−0.830499	−0.415250	−	0.909708i	$-0.636306\pi$
$457$	−8113.60	−0.830499	−0.415250	+	0.909708i	$0.636306\pi$
$458$	−7366.49	−0.751558
$459$	0	0
$460$	1170.55	0.118646
$461$	7870.77	0.795181	0.397590	−	0.917563i	$-0.369846\pi$
$461$	7870.77	0.795181	0.397590	+	0.917563i	$0.369846\pi$
$462$	0	0
$463$	966.227	0.0969857	0.0484928	−	0.998824i	$-0.484558\pi$
$463$	966.227	0.0969857	0.0484928	+	0.998824i	$0.484558\pi$
$464$	−1609.94	−0.161077
$465$	0	0
$466$	7257.52	0.721455
$467$	−1606.82	−0.159218	−0.0796092	−	0.996826i	$-0.525367\pi$
$467$	−1606.82	−0.159218	−0.0796092	+	0.996826i	$0.525367\pi$
$468$	0	0
$469$	3801.52	0.374281
$470$	−571.433	−0.0560813
$471$	0	0
$472$	−2287.94	−0.223116
$473$	4165.02	0.404879
$474$	0	0
$475$	−7199.86	−0.695479
$476$	1682.23	0.161985
$477$	0	0
$478$	556.461	0.0532467
$479$	−6081.41	−0.580097	−0.290049	−	0.957012i	$-0.593671\pi$
$479$	−6081.41	−0.580097	−0.290049	+	0.957012i	$0.593671\pi$
$480$	0	0
$481$	−17069.0	−1.61804
$482$	1561.54	0.147564
$483$	0	0
$484$	3016.55	0.283298
$485$	18220.3	1.70586
$486$	0	0
$487$	−3771.18	−0.350901	−0.175450	−	0.984488i	$-0.556138\pi$
$487$	−3771.18	−0.350901	−0.175450	+	0.984488i	$0.556138\pi$
$488$	2281.04	0.211594
$489$	0	0
$490$	−1613.07	−0.148716
$491$	−1836.86	−0.168831	−0.0844157	−	0.996431i	$-0.526902\pi$
$491$	−1836.86	−0.168831	−0.0844157	+	0.996431i	$0.526902\pi$
$492$	0	0
$493$	1444.45	0.131957
$494$	9347.69	0.851361
$495$	0	0
$496$	−6884.46	−0.623229
$497$	−9911.34	−0.894536
$498$	0	0
$499$	14874.8	1.33445	0.667224	−	0.744857i	$-0.267482\pi$
$499$	14874.8	1.33445	0.667224	+	0.744857i	$0.267482\pi$
$500$	1841.70	0.164726
$501$	0	0
$502$	−7372.03	−0.655438
$503$	−17495.3	−1.55085	−0.775423	−	0.631443i	$-0.782463\pi$
$503$	−17495.3	−1.55085	−0.775423	+	0.631443i	$0.782463\pi$
$504$	0	0
$505$	−17767.4	−1.56562
$506$	950.563	0.0835132
$507$	0	0
$508$	−3091.83	−0.270035
$509$	−140.279	−0.0122156	−0.00610780	−	0.999981i	$-0.501944\pi$
$509$	−140.279	−0.0122156	−0.00610780	+	0.999981i	$0.501944\pi$
$510$	0	0
$511$	20305.9	1.75788
$512$	11331.0	0.978052
$513$	0	0
$514$	−5343.17	−0.458516
$515$	1359.75	0.116345
$516$	0	0
$517$	220.775	0.0187808
$518$	−18958.4	−1.60808
$519$	0	0
$520$	14437.9	1.21758
$521$	4850.08	0.407842	0.203921	−	0.978987i	$-0.434631\pi$
$521$	4850.08	0.407842	0.203921	+	0.978987i	$0.434631\pi$
$522$	0	0
$523$	−9230.88	−0.771775	−0.385887	−	0.922546i	$-0.626105\pi$
$523$	−9230.88	−0.771775	−0.385887	+	0.922546i	$0.626105\pi$
$524$	5882.32	0.490401
$525$	0	0
$526$	11802.9	0.978385
$527$	6176.81	0.510561
$528$	0	0
$529$	−11134.1	−0.915109
$530$	14736.7	1.20778
$531$	0	0
$532$	−4939.57	−0.402552
$533$	−6315.17	−0.513209
$534$	0	0
$535$	2106.03	0.170190
$536$	−4728.97	−0.381082
$537$	0	0
$538$	3383.50	0.271139
$539$	623.212	0.0498027
$540$	0	0
$541$	−12999.0	−1.03303	−0.516515	−	0.856278i	$-0.672771\pi$
$541$	−12999.0	−1.03303	−0.516515	+	0.856278i	$0.672771\pi$
$542$	−10404.5	−0.824557
$543$	0	0
$544$	−3675.12	−0.289650
$545$	5630.04	0.442503
$546$	0	0
$547$	10145.9	0.793064	0.396532	−	0.918021i	$-0.370214\pi$
$547$	10145.9	0.793064	0.396532	+	0.918021i	$0.370214\pi$
$548$	−2021.20	−0.157558
$549$	0	0
$550$	−2201.57	−0.170682
$551$	−4241.38	−0.327929
$552$	0	0
$553$	15280.6	1.17504
$554$	7476.84	0.573394
$555$	0	0
$556$	4967.72	0.378918
$557$	−21192.0	−1.61209	−0.806045	−	0.591855i	$-0.798396\pi$
$557$	−21192.0	−1.61209	−0.806045	+	0.591855i	$0.798396\pi$
$558$	0	0
$559$	13608.7	1.02967
$560$	10266.6	0.774721
$561$	0	0
$562$	17138.6	1.28638
$563$	−7951.74	−0.595250	−0.297625	−	0.954683i	$-0.596195\pi$
$563$	−7951.74	−0.595250	−0.297625	+	0.954683i	$0.596195\pi$
$564$	0	0
$565$	−9024.76	−0.671990
$566$	17482.0	1.29827
$567$	0	0
$568$	12329.4	0.910791
$569$	7898.25	0.581919	0.290959	−	0.956735i	$-0.406026\pi$
$569$	7898.25	0.581919	0.290959	+	0.956735i	$0.406026\pi$
$570$	0	0
$571$	−9731.81	−0.713246	−0.356623	−	0.934248i	$-0.616072\pi$
$571$	−9731.81	−0.713246	−0.356623	+	0.934248i	$0.616072\pi$
$572$	−1359.89	−0.0994054
$573$	0	0
$574$	−7014.20	−0.510047
$575$	−2392.21	−0.173499
$576$	0	0
$577$	12959.2	0.935010	0.467505	−	0.883991i	$-0.345153\pi$
$577$	12959.2	0.935010	0.467505	+	0.883991i	$0.345153\pi$
$578$	8912.32	0.641356
$579$	0	0
$580$	−1597.07	−0.114335
$581$	−25110.5	−1.79305
$582$	0	0
$583$	−5693.56	−0.404465
$584$	−25259.8	−1.78983
$585$	0	0
$586$	12295.0	0.866724
$587$	−18175.4	−1.27799	−0.638994	−	0.769212i	$-0.720649\pi$
$587$	−18175.4	−1.27799	−0.638994	+	0.769212i	$0.720649\pi$
$588$	0	0
$589$	−18137.1	−1.26880
$590$	3054.19	0.213117
$591$	0	0
$592$	15098.8	1.04823
$593$	17260.7	1.19530	0.597649	−	0.801758i	$-0.296101\pi$
$593$	17260.7	1.19530	0.597649	+	0.801758i	$0.296101\pi$
$594$	0	0
$595$	−9211.31	−0.634667
$596$	−7386.96	−0.507687
$597$	0	0
$598$	3105.84	0.212387
$599$	−6861.91	−0.468064	−0.234032	−	0.972229i	$-0.575192\pi$
$599$	−6861.91	−0.468064	−0.234032	+	0.972229i	$0.575192\pi$
$600$	0	0
$601$	−10703.7	−0.726481	−0.363240	−	0.931695i	$-0.618330\pi$
$601$	−10703.7	−0.726481	−0.363240	+	0.931695i	$0.618330\pi$
$602$	15115.0	1.02333
$603$	0	0
$604$	5468.52	0.368396
$605$	−16517.6	−1.10997
$606$	0	0
$607$	−20024.6	−1.33900	−0.669500	−	0.742812i	$-0.733492\pi$
$607$	−20024.6	−1.33900	−0.669500	+	0.742812i	$0.733492\pi$
$608$	10791.3	0.719812
$609$	0	0
$610$	−3044.98	−0.202111
$611$	721.352	0.0477623
$612$	0	0
$613$	−255.865	−0.0168585	−0.00842927	−	0.999964i	$-0.502683\pi$
$613$	−255.865	−0.0168585	−0.00842927	+	0.999964i	$0.502683\pi$
$614$	−7508.56	−0.493519
$615$	0	0
$616$	−6195.56	−0.405237
$617$	9384.64	0.612336	0.306168	−	0.951978i	$-0.400953\pi$
$617$	9384.64	0.612336	0.306168	+	0.951978i	$0.400953\pi$
$618$	0	0
$619$	−18743.6	−1.21707	−0.608537	−	0.793525i	$-0.708243\pi$
$619$	−18743.6	−1.21707	−0.608537	+	0.793525i	$0.708243\pi$
$620$	−6829.41	−0.442380
$621$	0	0
$622$	20089.9	1.29506
$623$	−19391.0	−1.24701
$624$	0	0
$625$	−19388.8	−1.24088
$626$	22236.5	1.41973
$627$	0	0
$628$	3352.32	0.213013
$629$	−13546.7	−0.858735
$630$	0	0
$631$	−11378.7	−0.717873	−0.358936	−	0.933362i	$-0.616860\pi$
$631$	−11378.7	−0.717873	−0.358936	+	0.933362i	$0.616860\pi$
$632$	−19008.5	−1.19639
$633$	0	0
$634$	19938.4	1.24898
$635$	16929.8	1.05801
$636$	0	0
$637$	2036.26	0.126656
$638$	−1296.93	−0.0804792
$639$	0	0
$640$	−5594.40	−0.345528
$641$	−16950.9	−1.04449	−0.522247	−	0.852794i	$-0.674906\pi$
$641$	−16950.9	−1.04449	−0.522247	+	0.852794i	$0.674906\pi$
$642$	0	0
$643$	26137.9	1.60308	0.801539	−	0.597942i	$-0.204015\pi$
$643$	26137.9	1.60308	0.801539	+	0.597942i	$0.204015\pi$
$644$	−1641.21	−0.100424
$645$	0	0
$646$	7418.76	0.451838
$647$	−6604.26	−0.401299	−0.200649	−	0.979663i	$-0.564305\pi$
$647$	−6604.26	−0.401299	−0.200649	+	0.979663i	$0.564305\pi$
$648$	0	0
$649$	−1179.99	−0.0713696
$650$	−7193.33	−0.434070
$651$	0	0
$652$	1402.94	0.0842689
$653$	26726.0	1.60164	0.800818	−	0.598907i	$-0.204398\pi$
$653$	26726.0	1.60164	0.800818	+	0.598907i	$0.204398\pi$
$654$	0	0
$655$	−32209.5	−1.92142
$656$	5586.22	0.332477
$657$	0	0
$658$	801.199	0.0474681
$659$	15287.9	0.903692	0.451846	−	0.892096i	$-0.350766\pi$
$659$	15287.9	0.903692	0.451846	+	0.892096i	$0.350766\pi$
$660$	0	0
$661$	−17158.2	−1.00965	−0.504823	−	0.863223i	$-0.668442\pi$
$661$	−17158.2	−1.00965	−0.504823	+	0.863223i	$0.668442\pi$
$662$	15528.1	0.911656
$663$	0	0
$664$	31236.7	1.82563
$665$	27047.4	1.57722
$666$	0	0
$667$	−1409.23	−0.0818075
$668$	−2964.40	−0.171701
$669$	0	0
$670$	6312.75	0.364004
$671$	1176.44	0.0676838
$672$	0	0
$673$	26333.4	1.50829	0.754144	−	0.656709i	$-0.228052\pi$
$673$	26333.4	1.50829	0.754144	+	0.656709i	$0.228052\pi$
$674$	17306.3	0.989039
$675$	0	0
$676$	1222.97	0.0695820
$677$	−16592.7	−0.941961	−0.470980	−	0.882144i	$-0.656100\pi$
$677$	−16592.7	−0.941961	−0.470980	+	0.882144i	$0.656100\pi$
$678$	0	0
$679$	−25546.5	−1.44386
$680$	11458.6	0.646200
$681$	0	0
$682$	−5545.94	−0.311386
$683$	9583.36	0.536892	0.268446	−	0.963295i	$-0.413490\pi$
$683$	9583.36	0.536892	0.268446	+	0.963295i	$0.413490\pi$
$684$	0	0
$685$	11067.4	0.617319
$686$	−13551.0	−0.754200
$687$	0	0
$688$	−12037.8	−0.667061
$689$	−18603.0	−1.02862
$690$	0	0
$691$	14579.8	0.802663	0.401332	−	0.915933i	$-0.368547\pi$
$691$	14579.8	0.802663	0.401332	+	0.915933i	$0.368547\pi$
$692$	2682.24	0.147346
$693$	0	0
$694$	20233.8	1.10672
$695$	−27201.5	−1.48462
$696$	0	0
$697$	−5012.01	−0.272372
$698$	−22601.2	−1.22560
$699$	0	0
$700$	3801.15	0.205243
$701$	−18071.8	−0.973696	−0.486848	−	0.873487i	$-0.661854\pi$
$701$	−18071.8	−0.973696	−0.486848	+	0.873487i	$0.661854\pi$
$702$	0	0
$703$	39777.6	2.13405
$704$	7031.08	0.376412
$705$	0	0
$706$	10715.0	0.571195
$707$	24911.5	1.32517
$708$	0	0
$709$	36534.5	1.93523	0.967617	−	0.252424i	$-0.0812278\pi$
$709$	36534.5	1.93523	0.967617	+	0.252424i	$0.0812278\pi$
$710$	−16458.6	−0.869974
$711$	0	0
$712$	24121.8	1.26967
$713$	−6026.18	−0.316525
$714$	0	0
$715$	7446.28	0.389476
$716$	−985.315	−0.0514287
$717$	0	0
$718$	−4566.79	−0.237369
$719$	21135.3	1.09626	0.548131	−	0.836392i	$-0.315339\pi$
$719$	21135.3	1.09626	0.548131	+	0.836392i	$0.315339\pi$
$720$	0	0
$721$	−1906.49	−0.0984762
$722$	−5814.14	−0.299695
$723$	0	0
$724$	1939.29	0.0995484
$725$	3263.87	0.167196
$726$	0	0
$727$	27512.3	1.40354	0.701771	−	0.712403i	$-0.252393\pi$
$727$	27512.3	1.40354	0.701771	+	0.712403i	$0.252393\pi$
$728$	−20243.2	−1.03058
$729$	0	0
$730$	33719.6	1.70962
$731$	10800.5	0.546470
$732$	0	0
$733$	−3632.55	−0.183044	−0.0915219	−	0.995803i	$-0.529173\pi$
$733$	−3632.55	−0.183044	−0.0915219	+	0.995803i	$0.529173\pi$
$734$	17212.9	0.865587
$735$	0	0
$736$	3585.50	0.179570
$737$	−2438.95	−0.121899
$738$	0	0
$739$	−971.973	−0.0483824	−0.0241912	−	0.999707i	$-0.507701\pi$
$739$	−971.973	−0.0483824	−0.0241912	+	0.999707i	$0.507701\pi$
$740$	14978.0	0.744058
$741$	0	0
$742$	−20662.1	−1.02228
$743$	−22023.9	−1.08745	−0.543726	−	0.839263i	$-0.682987\pi$
$743$	−22023.9	−1.08745	−0.543726	+	0.839263i	$0.682987\pi$
$744$	0	0
$745$	40448.4	1.98915
$746$	22011.3	1.08028
$747$	0	0
$748$	−1079.27	−0.0527568
$749$	−2952.84	−0.144051
$750$	0	0
$751$	−23264.5	−1.13040	−0.565202	−	0.824952i	$-0.691202\pi$
$751$	−23264.5	−1.13040	−0.565202	+	0.824952i	$0.691202\pi$
$752$	−638.087	−0.0309424
$753$	0	0
$754$	−4237.53	−0.204671
$755$	−29943.7	−1.44339
$756$	0	0
$757$	9005.23	0.432366	0.216183	−	0.976353i	$-0.430639\pi$
$757$	9005.23	0.432366	0.216183	+	0.976353i	$0.430639\pi$
$758$	2231.11	0.106910
$759$	0	0
$760$	−33646.0	−1.60588
$761$	24788.3	1.18078	0.590391	−	0.807117i	$-0.298974\pi$
$761$	24788.3	1.18078	0.590391	+	0.807117i	$0.298974\pi$
$762$	0	0
$763$	−7893.81	−0.374541
$764$	−3548.62	−0.168043
$765$	0	0
$766$	7372.99	0.347777
$767$	−3855.48	−0.181504
$768$	0	0
$769$	−2584.28	−0.121186	−0.0605928	−	0.998163i	$-0.519299\pi$
$769$	−2584.28	−0.121186	−0.0605928	+	0.998163i	$0.519299\pi$
$770$	8270.52	0.387077
$771$	0	0
$772$	−5953.08	−0.277534
$773$	34381.6	1.59977	0.799883	−	0.600155i	$-0.204895\pi$
$773$	34381.6	1.59977	0.799883	+	0.600155i	$0.204895\pi$
$774$	0	0
$775$	13957.0	0.646905
$776$	31779.0	1.47010
$777$	0	0
$778$	12199.4	0.562174
$779$	14716.9	0.676876
$780$	0	0
$781$	6358.83	0.291341
$782$	2464.94	0.112719
$783$	0	0
$784$	−1801.22	−0.0820528
$785$	−18356.1	−0.834597
$786$	0	0
$787$	−30660.2	−1.38871	−0.694357	−	0.719631i	$-0.744311\pi$
$787$	−30660.2	−1.38871	−0.694357	+	0.719631i	$0.744311\pi$
$788$	−4752.93	−0.214868
$789$	0	0
$790$	25374.7	1.14277
$791$	12653.5	0.568783
$792$	0	0
$793$	3843.85	0.172130
$794$	14379.3	0.642696
$795$	0	0
$796$	−10054.3	−0.447694
$797$	41911.2	1.86270	0.931350	−	0.364126i	$-0.118632\pi$
$797$	41911.2	1.86270	0.931350	+	0.364126i	$0.118632\pi$
$798$	0	0
$799$	572.498	0.0253486
$800$	−8304.25	−0.367000
$801$	0	0
$802$	5054.76	0.222556
$803$	−13027.7	−0.572524
$804$	0	0
$805$	8986.69	0.393465
$806$	−18120.6	−0.791901
$807$	0	0
$808$	−30989.0	−1.34925
$809$	9024.04	0.392173	0.196087	−	0.980587i	$-0.437177\pi$
$809$	9024.04	0.392173	0.196087	+	0.980587i	$0.437177\pi$
$810$	0	0
$811$	−27300.3	−1.18205	−0.591026	−	0.806653i	$-0.701277\pi$
$811$	−27300.3	−1.18205	−0.591026	+	0.806653i	$0.701277\pi$
$812$	2239.23	0.0967752
$813$	0	0
$814$	12163.2	0.523733
$815$	−7682.00	−0.330170
$816$	0	0
$817$	−31713.6	−1.35804
$818$	8967.52	0.383303
$819$	0	0
$820$	5541.55	0.235999
$821$	−25623.8	−1.08926	−0.544628	−	0.838678i	$-0.683329\pi$
$821$	−25623.8	−1.08926	−0.544628	+	0.838678i	$0.683329\pi$
$822$	0	0
$823$	34934.6	1.47964	0.739820	−	0.672805i	$-0.234911\pi$
$823$	34934.6	1.47964	0.739820	+	0.672805i	$0.234911\pi$
$824$	2371.61	0.100266
$825$	0	0
$826$	−4282.24	−0.180385
$827$	43194.0	1.81621	0.908104	−	0.418745i	$-0.137530\pi$
$827$	43194.0	1.81621	0.908104	+	0.418745i	$0.137530\pi$
$828$	0	0
$829$	7297.54	0.305735	0.152867	−	0.988247i	$-0.451149\pi$
$829$	7297.54	0.305735	0.152867	+	0.988247i	$0.451149\pi$
$830$	−41698.2	−1.74381
$831$	0	0
$832$	22973.1	0.957272
$833$	1616.07	0.0672193
$834$	0	0
$835$	16232.0	0.672732
$836$	3169.09	0.131107
$837$	0	0
$838$	11579.9	0.477353
$839$	−19827.4	−0.815872	−0.407936	−	0.913010i	$-0.633751\pi$
$839$	−19827.4	−0.815872	−0.407936	+	0.913010i	$0.633751\pi$
$840$	0	0
$841$	−22466.3	−0.921165
$842$	9446.67	0.386644
$843$	0	0
$844$	−4372.26	−0.178317
$845$	−6696.57	−0.272626
$846$	0	0
$847$	23159.1	0.939499
$848$	16455.7	0.666379
$849$	0	0
$850$	−5708.96	−0.230372
$851$	13216.4	0.532376
$852$	0	0
$853$	−37573.2	−1.50819	−0.754093	−	0.656767i	$-0.771923\pi$
$853$	−37573.2	−1.50819	−0.754093	+	0.656767i	$0.771923\pi$
$854$	4269.33	0.171070
$855$	0	0
$856$	3673.24	0.146669
$857$	−26801.7	−1.06829	−0.534147	−	0.845391i	$-0.679367\pi$
$857$	−26801.7	−1.06829	−0.534147	+	0.845391i	$0.679367\pi$
$858$	0	0
$859$	43330.9	1.72111	0.860553	−	0.509361i	$-0.170118\pi$
$859$	43330.9	1.72111	0.860553	+	0.509361i	$0.170118\pi$
$860$	−11941.6	−0.473494
$861$	0	0
$862$	−3481.57	−0.137567
$863$	−31344.9	−1.23638	−0.618188	−	0.786030i	$-0.712133\pi$
$863$	−31344.9	−1.23638	−0.618188	+	0.786030i	$0.712133\pi$
$864$	0	0
$865$	−14687.0	−0.577309
$866$	28101.5	1.10269
$867$	0	0
$868$	9575.44	0.374437
$869$	−9803.58	−0.382697
$870$	0	0
$871$	−7968.94	−0.310008
$872$	9819.64	0.381347
$873$	0	0
$874$	−7237.85	−0.280119
$875$	14139.3	0.546282
$876$	0	0
$877$	23533.7	0.906132	0.453066	−	0.891477i	$-0.350330\pi$
$877$	23533.7	0.906132	0.453066	+	0.891477i	$0.350330\pi$
$878$	−36503.6	−1.40312
$879$	0	0
$880$	−6586.77	−0.252318
$881$	−29137.1	−1.11425	−0.557126	−	0.830428i	$-0.688096\pi$
$881$	−29137.1	−1.11425	−0.557126	+	0.830428i	$0.688096\pi$
$882$	0	0
$883$	3962.11	0.151003	0.0755015	−	0.997146i	$-0.475944\pi$
$883$	3962.11	0.151003	0.0755015	+	0.997146i	$0.475944\pi$
$884$	−3526.38	−0.134169
$885$	0	0
$886$	−34762.4	−1.31813
$887$	33346.2	1.26229	0.631147	−	0.775663i	$-0.282584\pi$
$887$	33346.2	1.26229	0.631147	+	0.775663i	$0.282584\pi$
$888$	0	0
$889$	−23737.0	−0.895517
$890$	−32200.5	−1.21277
$891$	0	0
$892$	4241.71	0.159219
$893$	−1681.04	−0.0629942
$894$	0	0
$895$	5395.23	0.201500
$896$	7843.84	0.292460
$897$	0	0
$898$	27860.9	1.03533
$899$	8221.97	0.305026
$900$	0	0
$901$	−14764.2	−0.545911
$902$	4500.11	0.166117
$903$	0	0
$904$	−15740.5	−0.579118
$905$	−10618.9	−0.390036
$906$	0	0
$907$	44217.6	1.61876	0.809382	−	0.587282i	$-0.199802\pi$
$907$	44217.6	1.61876	0.809382	+	0.587282i	$0.199802\pi$
$908$	−16929.2	−0.618740
$909$	0	0
$910$	27022.8	0.984394
$911$	16739.1	0.608771	0.304386	−	0.952549i	$-0.401549\pi$
$911$	16739.1	0.608771	0.304386	+	0.952549i	$0.401549\pi$
$912$	0	0
$913$	16110.2	0.583976
$914$	18890.8	0.683645
$915$	0	0
$916$	−8159.98	−0.294338
$917$	45160.5	1.62632
$918$	0	0
$919$	3927.70	0.140982	0.0704912	−	0.997512i	$-0.477543\pi$
$919$	3927.70	0.140982	0.0704912	+	0.997512i	$0.477543\pi$
$920$	−11179.1	−0.400615
$921$	0	0
$922$	−18325.4	−0.654572
$923$	20776.7	0.740923
$924$	0	0
$925$	−30610.1	−1.08806
$926$	−2249.65	−0.0798360
$927$	0	0
$928$	−4891.97	−0.173046
$929$	17538.0	0.619379	0.309690	−	0.950838i	$-0.399775\pi$
$929$	17538.0	0.619379	0.309690	+	0.950838i	$0.399775\pi$
$930$	0	0
$931$	−4745.31	−0.167048
$932$	8039.27	0.282548
$933$	0	0
$934$	3741.15	0.131064
$935$	5909.72	0.206704
$936$	0	0
$937$	−40623.3	−1.41633	−0.708167	−	0.706045i	$-0.750478\pi$
$937$	−40623.3	−1.41633	−0.708167	+	0.706045i	$0.750478\pi$
$938$	−8851.03	−0.308098
$939$	0	0
$940$	−632.985	−0.0219635
$941$	55072.6	1.90788	0.953941	−	0.299996i	$-0.0969852\pi$
$941$	55072.6	1.90788	0.953941	+	0.299996i	$0.0969852\pi$
$942$	0	0
$943$	4889.79	0.168858
$944$	3410.45	0.117585
$945$	0	0
$946$	−9697.36	−0.333286
$947$	−30177.6	−1.03552	−0.517761	−	0.855525i	$-0.673234\pi$
$947$	−30177.6	−1.03552	−0.517761	+	0.855525i	$0.673234\pi$
$948$	0	0
$949$	−42566.2	−1.45601
$950$	16763.3	0.572500
$951$	0	0
$952$	−16065.9	−0.546953
$953$	37187.7	1.26404	0.632018	−	0.774954i	$-0.282227\pi$
$953$	37187.7	1.26404	0.632018	+	0.774954i	$0.282227\pi$
$954$	0	0
$955$	19431.0	0.658401
$956$	616.400	0.0208534
$957$	0	0
$958$	14159.3	0.477521
$959$	−15517.5	−0.522508
$960$	0	0
$961$	5368.00	0.180188
$962$	39741.5	1.33193
$963$	0	0
$964$	1729.74	0.0577917
$965$	32596.9	1.08739
$966$	0	0
$967$	−29678.8	−0.986977	−0.493489	−	0.869752i	$-0.664279\pi$
$967$	−29678.8	−0.986977	−0.493489	+	0.869752i	$0.664279\pi$
$968$	−28809.1	−0.956571
$969$	0	0
$970$	−42422.1	−1.40422
$971$	−28501.9	−0.941989	−0.470994	−	0.882136i	$-0.656105\pi$
$971$	−28501.9	−0.941989	−0.470994	+	0.882136i	$0.656105\pi$
$972$	0	0
$973$	38138.9	1.25661
$974$	8780.40	0.288852
$975$	0	0
$976$	−3400.16	−0.111513
$977$	5073.80	0.166146	0.0830732	−	0.996543i	$-0.473526\pi$
$977$	5073.80	0.166146	0.0830732	+	0.996543i	$0.473526\pi$
$978$	0	0
$979$	12440.7	0.406137
$980$	−1786.82	−0.0582427
$981$	0	0
$982$	4276.73	0.138978
$983$	38507.8	1.24945	0.624724	−	0.780845i	$-0.285211\pi$
$983$	38507.8	1.24945	0.624724	+	0.780845i	$0.285211\pi$
$984$	0	0
$985$	26025.4	0.841865
$986$	−3363.10	−0.108624
$987$	0	0
$988$	10354.6	0.333424
$989$	−10537.1	−0.338786
$990$	0	0
$991$	−27625.7	−0.885529	−0.442765	−	0.896638i	$-0.646002\pi$
$991$	−27625.7	−0.885529	−0.442765	+	0.896638i	$0.646002\pi$
$992$	−20919.1	−0.669540
$993$	0	0
$994$	23076.4	0.736359
$995$	55053.7	1.75409
$996$	0	0
$997$	4766.12	0.151399	0.0756994	−	0.997131i	$-0.475881\pi$
$997$	4766.12	0.151399	0.0756994	+	0.997131i	$0.475881\pi$
$998$	−34632.9	−1.09848
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.a.1.6		22
3.2	odd	2		239.4.a.a.1.17	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.a.1.17	✓	22	3.2	odd	2
2151.4.a.a.1.6		22	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-2.32829 q^{2} -2.57908 q^{4} +14.1221 q^{5} -19.8005 q^{7} +24.6311 q^{8} +O(q^{10})\)	\(q-2.32829 q^{2} -2.57908 q^{4} +14.1221 q^{5} -19.8005 q^{7} +24.6311 q^{8} -32.8804 q^{10} +12.7034 q^{11} +41.5067 q^{13} +46.1012 q^{14} -36.7157 q^{16} +32.9417 q^{17} -96.7273 q^{19} -36.4221 q^{20} -29.5772 q^{22} -32.1384 q^{23} +74.4346 q^{25} -96.6396 q^{26} +51.0670 q^{28} +43.8488 q^{29} +187.507 q^{31} -111.564 q^{32} -76.6977 q^{34} -279.625 q^{35} -411.234 q^{37} +225.209 q^{38} +347.844 q^{40} -152.148 q^{41} +327.866 q^{43} -32.7631 q^{44} +74.8273 q^{46} +17.3791 q^{47} +49.0586 q^{49} -173.305 q^{50} -107.049 q^{52} -448.191 q^{53} +179.399 q^{55} -487.708 q^{56} -102.093 q^{58} -92.8880 q^{59} +92.6079 q^{61} -436.571 q^{62} +553.479 q^{64} +586.164 q^{65} -191.991 q^{67} -84.9592 q^{68} +651.047 q^{70} +500.561 q^{71} -1025.52 q^{73} +957.471 q^{74} +249.468 q^{76} -251.534 q^{77} -771.727 q^{79} -518.504 q^{80} +354.244 q^{82} +1268.18 q^{83} +465.207 q^{85} -763.367 q^{86} +312.900 q^{88} +979.322 q^{89} -821.853 q^{91} +82.8874 q^{92} -40.4636 q^{94} -1366.00 q^{95} +1290.20 q^{97} -114.223 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more