LMFDB - Embedded newform 2151.4.a.d.1.13

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:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q-2.06669 q^{2} -3.72878 q^{4} +0.727240 q^{5} +31.6777 q^{7} +24.2398 q^{8} +O(q^{10})$	\(q-2.06669 q^{2} -3.72878 q^{4} +0.727240 q^{5} +31.6777 q^{7} +24.2398 q^{8} -1.50298 q^{10} -37.9038 q^{11} +6.40766 q^{13} -65.4680 q^{14} -20.2660 q^{16} +70.3212 q^{17} +78.2130 q^{19} -2.71171 q^{20} +78.3356 q^{22} +26.8578 q^{23} -124.471 q^{25} -13.2427 q^{26} -118.119 q^{28} -216.912 q^{29} +31.8397 q^{31} -152.035 q^{32} -145.332 q^{34} +23.0373 q^{35} -199.966 q^{37} -161.642 q^{38} +17.6281 q^{40} -410.190 q^{41} +143.868 q^{43} +141.335 q^{44} -55.5068 q^{46} -42.6721 q^{47} +660.475 q^{49} +257.244 q^{50} -23.8927 q^{52} +206.555 q^{53} -27.5652 q^{55} +767.860 q^{56} +448.291 q^{58} -811.868 q^{59} -253.690 q^{61} -65.8029 q^{62} +476.337 q^{64} +4.65990 q^{65} +89.0387 q^{67} -262.212 q^{68} -47.6110 q^{70} -694.822 q^{71} +630.877 q^{73} +413.269 q^{74} -291.639 q^{76} -1200.70 q^{77} -974.426 q^{79} -14.7382 q^{80} +847.737 q^{82} -202.275 q^{83} +51.1404 q^{85} -297.331 q^{86} -918.781 q^{88} -297.884 q^{89} +202.980 q^{91} -100.147 q^{92} +88.1901 q^{94} +56.8796 q^{95} +2.55522 q^{97} -1365.00 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * 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 $

$2$	−2.06669	−0.730687	−0.365343	−	0.930873i	$-0.619048\pi$
$2$	−2.06669	−0.730687	−0.365343	+	0.930873i	$0.619048\pi$
$3$	0	0
$3$	0	0
$4$	−3.72878	−0.466097
$5$	0.727240	0.0650463	0.0325231	−	0.999471i	$-0.489646\pi$
$5$	0.727240	0.0650463	0.0325231	+	0.999471i	$0.489646\pi$
$6$	0	0
$7$	31.6777	1.71043	0.855217	−	0.518271i	$-0.173424\pi$
$7$	31.6777	1.71043	0.855217	+	0.518271i	$0.173424\pi$
$8$	24.2398	1.07126
$9$	0	0
$10$	−1.50298	−0.0475285
$11$	−37.9038	−1.03895	−0.519474	−	0.854486i	$-0.673872\pi$
$11$	−37.9038	−1.03895	−0.519474	+	0.854486i	$0.673872\pi$
$12$	0	0
$13$	6.40766	0.136705	0.0683525	−	0.997661i	$-0.478226\pi$
$13$	6.40766	0.136705	0.0683525	+	0.997661i	$0.478226\pi$
$14$	−65.4680	−1.24979
$15$	0	0
$16$	−20.2660	−0.316656
$17$	70.3212	1.00326	0.501629	−	0.865083i	$-0.332734\pi$
$17$	70.3212	1.00326	0.501629	+	0.865083i	$0.332734\pi$
$18$	0	0
$19$	78.2130	0.944384	0.472192	−	0.881496i	$-0.343463\pi$
$19$	78.2130	0.944384	0.472192	+	0.881496i	$0.343463\pi$
$20$	−2.71171	−0.0303179
$21$	0	0
$22$	78.3356	0.759146
$23$	26.8578	0.243488	0.121744	−	0.992562i	$-0.461151\pi$
$23$	26.8578	0.243488	0.121744	+	0.992562i	$0.461151\pi$
$24$	0	0
$25$	−124.471	−0.995769
$26$	−13.2427	−0.0998885
$27$	0	0
$28$	−118.119	−0.797228
$29$	−216.912	−1.38895	−0.694475	−	0.719516i	$-0.744363\pi$
$29$	−216.912	−1.38895	−0.694475	+	0.719516i	$0.744363\pi$
$30$	0	0
$31$	31.8397	0.184470	0.0922352	−	0.995737i	$-0.470599\pi$
$31$	31.8397	0.184470	0.0922352	+	0.995737i	$0.470599\pi$
$32$	−152.035	−0.839881
$33$	0	0
$34$	−145.332	−0.733068
$35$	23.0373	0.111257
$36$	0	0
$37$	−199.966	−0.888493	−0.444247	−	0.895905i	$-0.646529\pi$
$37$	−199.966	−0.888493	−0.444247	+	0.895905i	$0.646529\pi$
$38$	−161.642	−0.690049
$39$	0	0
$40$	17.6281	0.0696813
$41$	−410.190	−1.56246	−0.781230	−	0.624243i	$-0.785408\pi$
$41$	−410.190	−1.56246	−0.781230	+	0.624243i	$0.785408\pi$
$42$	0	0
$43$	143.868	0.510225	0.255112	−	0.966911i	$-0.417888\pi$
$43$	143.868	0.510225	0.255112	+	0.966911i	$0.417888\pi$
$44$	141.335	0.484251
$45$	0	0
$46$	−55.5068	−0.177914
$47$	−42.6721	−0.132433	−0.0662166	−	0.997805i	$-0.521093\pi$
$47$	−42.6721	−0.132433	−0.0662166	+	0.997805i	$0.521093\pi$
$48$	0	0
$49$	660.475	1.92558
$50$	257.244	0.727595
$51$	0	0
$52$	−23.8927	−0.0637178
$53$	206.555	0.535329	0.267665	−	0.963512i	$-0.413748\pi$
$53$	206.555	0.535329	0.267665	+	0.963512i	$0.413748\pi$
$54$	0	0
$55$	−27.5652	−0.0675797
$56$	767.860	1.83231
$57$	0	0
$58$	448.291	1.01489
$59$	−811.868	−1.79146	−0.895730	−	0.444598i	$-0.853347\pi$
$59$	−811.868	−1.79146	−0.895730	+	0.444598i	$0.853347\pi$
$60$	0	0
$61$	−253.690	−0.532486	−0.266243	−	0.963906i	$-0.585782\pi$
$61$	−253.690	−0.532486	−0.266243	+	0.963906i	$0.585782\pi$
$62$	−65.8029	−0.134790
$63$	0	0
$64$	476.337	0.930346
$65$	4.65990	0.00889215
$66$	0	0
$67$	89.0387	0.162355	0.0811777	−	0.996700i	$-0.474132\pi$
$67$	89.0387	0.162355	0.0811777	+	0.996700i	$0.474132\pi$
$68$	−262.212	−0.467616
$69$	0	0
$70$	−47.6110	−0.0812942
$71$	−694.822	−1.16141	−0.580705	−	0.814114i	$-0.697223\pi$
$71$	−694.822	−1.16141	−0.580705	+	0.814114i	$0.697223\pi$
$72$	0	0
$73$	630.877	1.01149	0.505744	−	0.862684i	$-0.331218\pi$
$73$	630.877	1.01149	0.505744	+	0.862684i	$0.331218\pi$
$74$	413.269	0.649210
$75$	0	0
$76$	−291.639	−0.440175
$77$	−1200.70	−1.77705
$78$	0	0
$79$	−974.426	−1.38774	−0.693870	−	0.720100i	$-0.744096\pi$
$79$	−974.426	−1.38774	−0.693870	+	0.720100i	$0.744096\pi$
$80$	−14.7382	−0.0205973
$81$	0	0
$82$	847.737	1.14167
$83$	−202.275	−0.267501	−0.133750	−	0.991015i	$-0.542702\pi$
$83$	−202.275	−0.267501	−0.133750	+	0.991015i	$0.542702\pi$
$84$	0	0
$85$	51.1404	0.0652583
$86$	−297.331	−0.372814
$87$	0	0
$88$	−918.781	−1.11298
$89$	−297.884	−0.354783	−0.177391	−	0.984140i	$-0.556766\pi$
$89$	−297.884	−0.354783	−0.177391	+	0.984140i	$0.556766\pi$
$90$	0	0
$91$	202.980	0.233825
$92$	−100.147	−0.113489
$93$	0	0
$94$	88.1901	0.0967672
$95$	56.8796	0.0614287
$96$	0	0
$97$	2.55522	0.00267467	0.00133734	−	0.999999i	$-0.499574\pi$
$97$	2.55522	0.00267467	0.00133734	+	0.999999i	$0.499574\pi$
$98$	−1365.00	−1.40700
$99$	0	0
$100$	464.125	0.464125
$101$	−133.653	−0.131673	−0.0658366	−	0.997830i	$-0.520972\pi$
$101$	−133.653	−0.131673	−0.0658366	+	0.997830i	$0.520972\pi$
$102$	0	0
$103$	−42.7938	−0.0409378	−0.0204689	−	0.999790i	$-0.506516\pi$
$103$	−42.7938	−0.0409378	−0.0204689	+	0.999790i	$0.506516\pi$
$104$	155.320	0.146446
$105$	0	0
$106$	−426.885	−0.391158
$107$	197.511	0.178450	0.0892248	−	0.996012i	$-0.471561\pi$
$107$	197.511	0.178450	0.0892248	+	0.996012i	$0.471561\pi$
$108$	0	0
$109$	1296.76	1.13951	0.569756	−	0.821814i	$-0.307038\pi$
$109$	1296.76	1.13951	0.569756	+	0.821814i	$0.307038\pi$
$110$	56.9687	0.0493796
$111$	0	0
$112$	−641.980	−0.541620
$113$	−130.499	−0.108640	−0.0543200	−	0.998524i	$-0.517299\pi$
$113$	−130.499	−0.108640	−0.0543200	+	0.998524i	$0.517299\pi$
$114$	0	0
$115$	19.5320	0.0158380
$116$	808.817	0.647386
$117$	0	0
$118$	1677.88	1.30900
$119$	2227.61	1.71601
$120$	0	0
$121$	105.700	0.0794137
$122$	524.299	0.389080
$123$	0	0
$124$	−118.723	−0.0859811
$125$	−181.425	−0.129817
$126$	0	0
$127$	333.074	0.232720	0.116360	−	0.993207i	$-0.462877\pi$
$127$	333.074	0.232720	0.116360	+	0.993207i	$0.462877\pi$
$128$	231.834	0.160090
$129$	0	0
$130$	−9.63059	−0.00649737
$131$	2333.35	1.55622	0.778112	−	0.628125i	$-0.216178\pi$
$131$	2333.35	1.55622	0.778112	+	0.628125i	$0.216178\pi$
$132$	0	0
$133$	2477.61	1.61531
$134$	−184.016	−0.118631
$135$	0	0
$136$	1704.57	1.07475
$137$	−333.313	−0.207860	−0.103930	−	0.994585i	$-0.533142\pi$
$137$	−333.313	−0.207860	−0.103930	+	0.994585i	$0.533142\pi$
$138$	0	0
$139$	−1185.90	−0.723646	−0.361823	−	0.932247i	$-0.617845\pi$
$139$	−1185.90	−0.723646	−0.361823	+	0.932247i	$0.617845\pi$
$140$	−85.9008	−0.0518567
$141$	0	0
$142$	1435.98	0.848627
$143$	−242.875	−0.142029
$144$	0	0
$145$	−157.747	−0.0903461
$146$	−1303.83	−0.739080
$147$	0	0
$148$	745.630	0.414124
$149$	2216.56	1.21871	0.609355	−	0.792898i	$-0.291429\pi$
$149$	2216.56	1.21871	0.609355	+	0.792898i	$0.291429\pi$
$150$	0	0
$151$	−1224.53	−0.659941	−0.329970	−	0.943991i	$-0.607039\pi$
$151$	−1224.53	−0.659941	−0.329970	+	0.943991i	$0.607039\pi$
$152$	1895.87	1.01168
$153$	0	0
$154$	2481.49	1.29847
$155$	23.1551	0.0119991
$156$	0	0
$157$	−1107.62	−0.563045	−0.281523	−	0.959555i	$-0.590839\pi$
$157$	−1107.62	−0.563045	−0.281523	+	0.959555i	$0.590839\pi$
$158$	2013.84	1.01400
$159$	0	0
$160$	−110.566	−0.0546311
$161$	850.791	0.416470
$162$	0	0
$163$	−3031.55	−1.45674	−0.728372	−	0.685182i	$-0.759723\pi$
$163$	−3031.55	−1.45674	−0.728372	+	0.685182i	$0.759723\pi$
$164$	1529.51	0.728259
$165$	0	0
$166$	418.040	0.195459
$167$	1153.86	0.534662	0.267331	−	0.963605i	$-0.413858\pi$
$167$	1153.86	0.534662	0.267331	+	0.963605i	$0.413858\pi$
$168$	0	0
$169$	−2155.94	−0.981312
$170$	−105.691	−0.0476833
$171$	0	0
$172$	−536.451	−0.237814
$173$	−919.425	−0.404061	−0.202031	−	0.979379i	$-0.564754\pi$
$173$	−919.425	−0.404061	−0.202031	+	0.979379i	$0.564754\pi$
$174$	0	0
$175$	−3942.95	−1.70320
$176$	768.159	0.328990
$177$	0	0
$178$	615.636	0.259235
$179$	3355.05	1.40094	0.700469	−	0.713683i	$-0.252974\pi$
$179$	3355.05	1.40094	0.700469	+	0.713683i	$0.252974\pi$
$180$	0	0
$181$	611.329	0.251048	0.125524	−	0.992091i	$-0.459939\pi$
$181$	611.329	0.251048	0.125524	+	0.992091i	$0.459939\pi$
$182$	−419.497	−0.170853
$183$	0	0
$184$	651.026	0.260839
$185$	−145.423	−0.0577932
$186$	0	0
$187$	−2665.44	−1.04233
$188$	159.115	0.0617268
$189$	0	0
$190$	−117.553	−0.0448851
$191$	328.049	0.124276	0.0621381	−	0.998068i	$-0.480208\pi$
$191$	328.049	0.124276	0.0621381	+	0.998068i	$0.480208\pi$
$192$	0	0
$193$	5276.17	1.96781	0.983904	−	0.178696i	$-0.0571877\pi$
$193$	5276.17	1.96781	0.983904	+	0.178696i	$0.0571877\pi$
$194$	−5.28085	−0.00195435
$195$	0	0
$196$	−2462.76	−0.897508
$197$	2637.36	0.953829	0.476914	−	0.878950i	$-0.341755\pi$
$197$	2637.36	0.953829	0.476914	+	0.878950i	$0.341755\pi$
$198$	0	0
$199$	162.317	0.0578208	0.0289104	−	0.999582i	$-0.490796\pi$
$199$	162.317	0.0578208	0.0289104	+	0.999582i	$0.490796\pi$
$200$	−3017.15	−1.06672
$201$	0	0
$202$	276.220	0.0962118
$203$	−6871.27	−2.37571
$204$	0	0
$205$	−298.306	−0.101632
$206$	88.4416	0.0299127
$207$	0	0
$208$	−129.858	−0.0432885
$209$	−2964.57	−0.981166
$210$	0	0
$211$	699.003	0.228063	0.114032	−	0.993477i	$-0.463623\pi$
$211$	699.003	0.228063	0.114032	+	0.993477i	$0.463623\pi$
$212$	−770.196	−0.249515
$213$	0	0
$214$	−408.194	−0.130391
$215$	104.626	0.0331882
$216$	0	0
$217$	1008.61	0.315524
$218$	−2680.00	−0.832626
$219$	0	0
$220$	102.784	0.0314987
$221$	450.594	0.137150
$222$	0	0
$223$	−991.830	−0.297838	−0.148919	−	0.988849i	$-0.547579\pi$
$223$	−991.830	−0.297838	−0.148919	+	0.988849i	$0.547579\pi$
$224$	−4816.10	−1.43656
$225$	0	0
$226$	269.702	0.0793818
$227$	−4279.20	−1.25119	−0.625596	−	0.780148i	$-0.715144\pi$
$227$	−4279.20	−1.25119	−0.625596	+	0.780148i	$0.715144\pi$
$228$	0	0
$229$	−1508.39	−0.435270	−0.217635	−	0.976030i	$-0.569834\pi$
$229$	−1508.39	−0.435270	−0.217635	+	0.976030i	$0.569834\pi$
$230$	−40.3667	−0.0115726
$231$	0	0
$232$	−5257.90	−1.48792
$233$	−4506.39	−1.26705	−0.633526	−	0.773721i	$-0.718393\pi$
$233$	−4506.39	−1.26705	−0.633526	+	0.773721i	$0.718393\pi$
$234$	0	0
$235$	−31.0328	−0.00861429
$236$	3027.27	0.834995
$237$	0	0
$238$	−4603.79	−1.25386
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−3218.97	−0.860381	−0.430191	−	0.902738i	$-0.641554\pi$
$241$	−3218.97	−0.860381	−0.430191	+	0.902738i	$0.641554\pi$
$242$	−218.449	−0.0580265
$243$	0	0
$244$	945.952	0.248190
$245$	480.323	0.125252
$246$	0	0
$247$	501.162	0.129102
$248$	771.788	0.197615
$249$	0	0
$250$	374.951	0.0948558
$251$	1228.72	0.308989	0.154494	−	0.987994i	$-0.450625\pi$
$251$	1228.72	0.308989	0.154494	+	0.987994i	$0.450625\pi$
$252$	0	0
$253$	−1018.01	−0.252972
$254$	−688.361	−0.170046
$255$	0	0
$256$	−4289.83	−1.04732
$257$	1297.16	0.314842	0.157421	−	0.987532i	$-0.449682\pi$
$257$	1297.16	0.314842	0.157421	+	0.987532i	$0.449682\pi$
$258$	0	0
$259$	−6334.46	−1.51971
$260$	−17.3757	−0.00414460
$261$	0	0
$262$	−4822.31	−1.13711
$263$	−6079.71	−1.42544	−0.712721	−	0.701448i	$-0.752537\pi$
$263$	−6079.71	−1.42544	−0.712721	+	0.701448i	$0.752537\pi$
$264$	0	0
$265$	150.215	0.0348212
$266$	−5120.45	−1.18028
$267$	0	0
$268$	−332.006	−0.0756734
$269$	−4900.37	−1.11071	−0.555355	−	0.831613i	$-0.687418\pi$
$269$	−4900.37	−1.11071	−0.555355	+	0.831613i	$0.687418\pi$
$270$	0	0
$271$	−6183.67	−1.38609	−0.693046	−	0.720893i	$-0.743732\pi$
$271$	−6183.67	−1.38609	−0.693046	+	0.720893i	$0.743732\pi$
$272$	−1425.13	−0.317688
$273$	0	0
$274$	688.856	0.151881
$275$	4717.93	1.03455
$276$	0	0
$277$	2654.26	0.575738	0.287869	−	0.957670i	$-0.407053\pi$
$277$	2654.26	0.575738	0.287869	+	0.957670i	$0.407053\pi$
$278$	2450.89	0.528758
$279$	0	0
$280$	558.418	0.119185
$281$	3776.94	0.801828	0.400914	−	0.916116i	$-0.368693\pi$
$281$	3776.94	0.801828	0.400914	+	0.916116i	$0.368693\pi$
$282$	0	0
$283$	−1900.19	−0.399133	−0.199567	−	0.979884i	$-0.563953\pi$
$283$	−1900.19	−0.399133	−0.199567	+	0.979884i	$0.563953\pi$
$284$	2590.84	0.541330
$285$	0	0
$286$	501.948	0.103779
$287$	−12993.9	−2.67249
$288$	0	0
$289$	32.0735	0.00652828
$290$	326.015	0.0660147
$291$	0	0
$292$	−2352.40	−0.471451
$293$	3554.52	0.708728	0.354364	−	0.935108i	$-0.384697\pi$
$293$	3554.52	0.708728	0.354364	+	0.935108i	$0.384697\pi$
$294$	0	0
$295$	−590.422	−0.116528
$296$	−4847.14	−0.951805
$297$	0	0
$298$	−4580.95	−0.890495
$299$	172.095	0.0332860
$300$	0	0
$301$	4557.40	0.872705
$302$	2530.73	0.482210
$303$	0	0
$304$	−1585.07	−0.299045
$305$	−184.493	−0.0346362
$306$	0	0
$307$	−2814.43	−0.523218	−0.261609	−	0.965174i	$-0.584253\pi$
$307$	−2814.43	−0.523218	−0.261609	+	0.965174i	$0.584253\pi$
$308$	4477.16	0.828279
$309$	0	0
$310$	−47.8545	−0.00876759
$311$	8889.78	1.62088	0.810439	−	0.585822i	$-0.199228\pi$
$311$	8889.78	1.62088	0.810439	+	0.585822i	$0.199228\pi$
$312$	0	0
$313$	917.089	0.165613	0.0828066	−	0.996566i	$-0.473612\pi$
$313$	917.089	0.165613	0.0828066	+	0.996566i	$0.473612\pi$
$314$	2289.12	0.411409
$315$	0	0
$316$	3633.42	0.646822
$317$	−1993.09	−0.353133	−0.176567	−	0.984289i	$-0.556499\pi$
$317$	−1993.09	−0.353133	−0.176567	+	0.984289i	$0.556499\pi$
$318$	0	0
$319$	8221.80	1.44305
$320$	346.411	0.0605156
$321$	0	0
$322$	−1758.32	−0.304309
$323$	5500.04	0.947462
$324$	0	0
$325$	−797.568	−0.136127
$326$	6265.28	1.06442
$327$	0	0
$328$	−9942.92	−1.67380
$329$	−1351.75	−0.226518
$330$	0	0
$331$	−11428.9	−1.89786	−0.948930	−	0.315487i	$-0.897832\pi$
$331$	−11428.9	−1.89786	−0.948930	+	0.315487i	$0.897832\pi$
$332$	754.238	0.124681
$333$	0	0
$334$	−2384.68	−0.390670
$335$	64.7525	0.0105606
$336$	0	0
$337$	−8051.91	−1.30153	−0.650765	−	0.759279i	$-0.725552\pi$
$337$	−8051.91	−1.30153	−0.650765	+	0.759279i	$0.725552\pi$
$338$	4455.67	0.717031
$339$	0	0
$340$	−190.691	−0.0304167
$341$	−1206.85	−0.191655
$342$	0	0
$343$	10056.9	1.58315
$344$	3487.33	0.546582
$345$	0	0
$346$	1900.17	0.295242
$347$	636.918	0.0985347	0.0492673	−	0.998786i	$-0.484311\pi$
$347$	636.918	0.0985347	0.0492673	+	0.998786i	$0.484311\pi$
$348$	0	0
$349$	−2416.00	−0.370560	−0.185280	−	0.982686i	$-0.559319\pi$
$349$	−2416.00	−0.370560	−0.185280	+	0.982686i	$0.559319\pi$
$350$	8148.88	1.24450
$351$	0	0
$352$	5762.70	0.872593
$353$	−9568.72	−1.44275	−0.721376	−	0.692543i	$-0.756490\pi$
$353$	−9568.72	−1.44275	−0.721376	+	0.692543i	$0.756490\pi$
$354$	0	0
$355$	−505.302	−0.0755455
$356$	1110.74	0.165363
$357$	0	0
$358$	−6933.85	−1.02365
$359$	−7963.15	−1.17069	−0.585347	−	0.810783i	$-0.699042\pi$
$359$	−7963.15	−1.17069	−0.585347	+	0.810783i	$0.699042\pi$
$360$	0	0
$361$	−741.723	−0.108139
$362$	−1263.43	−0.183437
$363$	0	0
$364$	−756.866	−0.108985
$365$	458.799	0.0657935
$366$	0	0
$367$	−3277.67	−0.466194	−0.233097	−	0.972453i	$-0.574886\pi$
$367$	−3277.67	−0.466194	−0.233097	+	0.972453i	$0.574886\pi$
$368$	−544.299	−0.0771021
$369$	0	0
$370$	300.546	0.0422287
$371$	6543.17	0.915645
$372$	0	0
$373$	1750.01	0.242927	0.121464	−	0.992596i	$-0.461241\pi$
$373$	1750.01	0.242927	0.121464	+	0.992596i	$0.461241\pi$
$374$	5508.65	0.761620
$375$	0	0
$376$	−1034.36	−0.141870
$377$	−1389.90	−0.189876
$378$	0	0
$379$	−10252.3	−1.38952	−0.694759	−	0.719242i	$-0.744489\pi$
$379$	−10252.3	−1.38952	−0.694759	+	0.719242i	$0.744489\pi$
$380$	−212.091	−0.0286317
$381$	0	0
$382$	−677.976	−0.0908070
$383$	−8385.54	−1.11875	−0.559375	−	0.828915i	$-0.688959\pi$
$383$	−8385.54	−1.11875	−0.559375	+	0.828915i	$0.688959\pi$
$384$	0	0
$385$	−873.200	−0.115591
$386$	−10904.2	−1.43785
$387$	0	0
$388$	−9.52784	−0.00124666
$389$	15212.0	1.98272	0.991361	−	0.131161i	$-0.0418703\pi$
$389$	15212.0	1.98272	0.991361	+	0.131161i	$0.0418703\pi$
$390$	0	0
$391$	1888.67	0.244282
$392$	16009.8	2.06279
$393$	0	0
$394$	−5450.62	−0.696950
$395$	−708.641	−0.0902674
$396$	0	0
$397$	−9709.74	−1.22750	−0.613751	−	0.789500i	$-0.710340\pi$
$397$	−9709.74	−1.22750	−0.613751	+	0.789500i	$0.710340\pi$
$398$	−335.459	−0.0422489
$399$	0	0
$400$	2522.53	0.315317
$401$	−8435.06	−1.05044	−0.525220	−	0.850966i	$-0.676017\pi$
$401$	−8435.06	−1.05044	−0.525220	+	0.850966i	$0.676017\pi$
$402$	0	0
$403$	204.018	0.0252180
$404$	498.363	0.0613725
$405$	0	0
$406$	14200.8	1.73590
$407$	7579.49	0.923099
$408$	0	0
$409$	−7400.62	−0.894712	−0.447356	−	0.894356i	$-0.647634\pi$
$409$	−7400.62	−0.894712	−0.447356	+	0.894356i	$0.647634\pi$
$410$	616.508	0.0742613
$411$	0	0
$412$	159.568	0.0190810
$413$	−25718.1	−3.06417
$414$	0	0
$415$	−147.102	−0.0173999
$416$	−974.186	−0.114816
$417$	0	0
$418$	6126.86	0.716925
$419$	2628.11	0.306424	0.153212	−	0.988193i	$-0.451038\pi$
$419$	2628.11	0.306424	0.153212	+	0.988193i	$0.451038\pi$
$420$	0	0
$421$	8329.39	0.964250	0.482125	−	0.876102i	$-0.339865\pi$
$421$	8329.39	0.964250	0.482125	+	0.876102i	$0.339865\pi$
$422$	−1444.62	−0.166643
$423$	0	0
$424$	5006.84	0.573475
$425$	−8752.96	−0.999014
$426$	0	0
$427$	−8036.30	−0.910781
$428$	−736.474	−0.0831748
$429$	0	0
$430$	−216.231	−0.0242502
$431$	3365.79	0.376158	0.188079	−	0.982154i	$-0.439774\pi$
$431$	3365.79	0.376158	0.188079	+	0.982154i	$0.439774\pi$
$432$	0	0
$433$	7318.66	0.812268	0.406134	−	0.913813i	$-0.366877\pi$
$433$	7318.66	0.812268	0.406134	+	0.913813i	$0.366877\pi$
$434$	−2084.48	−0.230549
$435$	0	0
$436$	−4835.32	−0.531123
$437$	2100.63	0.229946
$438$	0	0
$439$	2340.45	0.254450	0.127225	−	0.991874i	$-0.459393\pi$
$439$	2340.45	0.254450	0.127225	+	0.991874i	$0.459393\pi$
$440$	−668.174	−0.0723953
$441$	0	0
$442$	−931.240	−0.100214
$443$	2676.98	0.287104	0.143552	−	0.989643i	$-0.454148\pi$
$443$	2676.98	0.287104	0.143552	+	0.989643i	$0.454148\pi$
$444$	0	0
$445$	−216.633	−0.0230773
$446$	2049.81	0.217626
$447$	0	0
$448$	15089.3	1.59129
$449$	−10800.8	−1.13523	−0.567617	−	0.823293i	$-0.692135\pi$
$449$	−10800.8	−1.13523	−0.567617	+	0.823293i	$0.692135\pi$
$450$	0	0
$451$	15547.8	1.62332
$452$	486.602	0.0506368
$453$	0	0
$454$	8843.79	0.914229
$455$	147.615	0.0152094
$456$	0	0
$457$	−10482.8	−1.07301	−0.536506	−	0.843896i	$-0.680256\pi$
$457$	−10482.8	−1.07301	−0.536506	+	0.843896i	$0.680256\pi$
$458$	3117.37	0.318046
$459$	0	0
$460$	−72.8306	−0.00738205
$461$	5130.78	0.518361	0.259180	−	0.965829i	$-0.416548\pi$
$461$	5130.78	0.518361	0.259180	+	0.965829i	$0.416548\pi$
$462$	0	0
$463$	−770.276	−0.0773169	−0.0386585	−	0.999252i	$-0.512308\pi$
$463$	−770.276	−0.0773169	−0.0386585	+	0.999252i	$0.512308\pi$
$464$	4395.94	0.439820
$465$	0	0
$466$	9313.32	0.925819
$467$	12660.1	1.25448	0.627238	−	0.778828i	$-0.284186\pi$
$467$	12660.1	1.25448	0.627238	+	0.778828i	$0.284186\pi$
$468$	0	0
$469$	2820.54	0.277698
$470$	64.1354	0.00629435
$471$	0	0
$472$	−19679.5	−1.91912
$473$	−5453.14	−0.530097
$474$	0	0
$475$	−9735.26	−0.940388
$476$	−8306.27	−0.799826
$477$	0	0
$478$	493.940	0.0472642
$479$	−3588.35	−0.342288	−0.171144	−	0.985246i	$-0.554746\pi$
$479$	−3588.35	−0.342288	−0.171144	+	0.985246i	$0.554746\pi$
$480$	0	0
$481$	−1281.32	−0.121461
$482$	6652.62	0.628669
$483$	0	0
$484$	−394.130	−0.0370145
$485$	1.85826	0.000173977 0
$486$	0	0
$487$	−4430.10	−0.412211	−0.206106	−	0.978530i	$-0.566079\pi$
$487$	−4430.10	−0.412211	−0.206106	+	0.978530i	$0.566079\pi$
$488$	−6149.38	−0.570429
$489$	0	0
$490$	−992.681	−0.0915199
$491$	2089.98	0.192097	0.0960484	−	0.995377i	$-0.469380\pi$
$491$	2089.98	0.192097	0.0960484	+	0.995377i	$0.469380\pi$
$492$	0	0
$493$	−15253.5	−1.39348
$494$	−1035.75	−0.0943331
$495$	0	0
$496$	−645.264	−0.0584137
$497$	−22010.3	−1.98652
$498$	0	0
$499$	−6046.56	−0.542447	−0.271223	−	0.962516i	$-0.587428\pi$
$499$	−6046.56	−0.542447	−0.271223	+	0.962516i	$0.587428\pi$
$500$	676.494	0.0605075
$501$	0	0
$502$	−2539.39	−0.225774
$503$	−7379.27	−0.654126	−0.327063	−	0.945003i	$-0.606059\pi$
$503$	−7379.27	−0.654126	−0.327063	+	0.945003i	$0.606059\pi$
$504$	0	0
$505$	−97.1979	−0.00856485
$506$	2103.92	0.184843
$507$	0	0
$508$	−1241.96	−0.108470
$509$	4423.59	0.385210	0.192605	−	0.981276i	$-0.438306\pi$
$509$	4423.59	0.385210	0.192605	+	0.981276i	$0.438306\pi$
$510$	0	0
$511$	19984.7	1.73008
$512$	7011.09	0.605174
$513$	0	0
$514$	−2680.82	−0.230051
$515$	−31.1213	−0.00266285
$516$	0	0
$517$	1617.44	0.137591
$518$	13091.4	1.11043
$519$	0	0
$520$	112.955	0.00952578
$521$	1502.78	0.126368	0.0631841	−	0.998002i	$-0.479874\pi$
$521$	1502.78	0.126368	0.0631841	+	0.998002i	$0.479874\pi$
$522$	0	0
$523$	21244.8	1.77623	0.888116	−	0.459619i	$-0.152014\pi$
$523$	21244.8	1.77623	0.888116	+	0.459619i	$0.152014\pi$
$524$	−8700.53	−0.725352
$525$	0	0
$526$	12564.9	1.04155
$527$	2239.01	0.185071
$528$	0	0
$529$	−11445.7	−0.940713
$530$	−310.448	−0.0254434
$531$	0	0
$532$	−9238.44	−0.752890
$533$	−2628.36	−0.213596
$534$	0	0
$535$	143.638	0.0116075
$536$	2158.28	0.173924
$537$	0	0
$538$	10127.6	0.811581
$539$	−25034.5	−2.00058
$540$	0	0
$541$	15504.6	1.23215	0.616075	−	0.787687i	$-0.288722\pi$
$541$	15504.6	1.23215	0.616075	+	0.787687i	$0.288722\pi$
$542$	12779.7	1.01280
$543$	0	0
$544$	−10691.3	−0.842618
$545$	943.053	0.0741211
$546$	0	0
$547$	−18970.3	−1.48284	−0.741419	−	0.671042i	$-0.765847\pi$
$547$	−18970.3	−1.48284	−0.741419	+	0.671042i	$0.765847\pi$
$548$	1242.85	0.0968831
$549$	0	0
$550$	−9750.52	−0.755934
$551$	−16965.4	−1.31170
$552$	0	0
$553$	−30867.5	−2.37364
$554$	−5485.55	−0.420684
$555$	0	0
$556$	4421.96	0.337289
$557$	−14427.0	−1.09748	−0.548738	−	0.835995i	$-0.684891\pi$
$557$	−14427.0	−1.09748	−0.548738	+	0.835995i	$0.684891\pi$
$558$	0	0
$559$	921.856	0.0697502
$560$	−466.873	−0.0352303
$561$	0	0
$562$	−7805.79	−0.585885
$563$	4663.66	0.349112	0.174556	−	0.984647i	$-0.444151\pi$
$563$	4663.66	0.349112	0.174556	+	0.984647i	$0.444151\pi$
$564$	0	0
$565$	−94.9041	−0.00706663
$566$	3927.11	0.291641
$567$	0	0
$568$	−16842.3	−1.24417
$569$	2708.51	0.199555	0.0997773	−	0.995010i	$-0.468187\pi$
$569$	2708.51	0.199555	0.0997773	+	0.995010i	$0.468187\pi$
$570$	0	0
$571$	24980.5	1.83083	0.915414	−	0.402513i	$-0.131863\pi$
$571$	24980.5	1.83083	0.915414	+	0.402513i	$0.131863\pi$
$572$	905.625	0.0661995
$573$	0	0
$574$	26854.3	1.95275
$575$	−3343.01	−0.242458
$576$	0	0
$577$	−4978.22	−0.359179	−0.179589	−	0.983742i	$-0.557477\pi$
$577$	−4978.22	−0.359179	−0.179589	+	0.983742i	$0.557477\pi$
$578$	−66.2860	−0.00477013
$579$	0	0
$580$	588.204	0.0421101
$581$	−6407.60	−0.457542
$582$	0	0
$583$	−7829.21	−0.556179
$584$	15292.3	1.08356
$585$	0	0
$586$	−7346.10	−0.517858
$587$	−10037.3	−0.705761	−0.352881	−	0.935668i	$-0.614798\pi$
$587$	−10037.3	−0.705761	−0.352881	+	0.935668i	$0.614798\pi$
$588$	0	0
$589$	2490.28	0.174211
$590$	1220.22	0.0851454
$591$	0	0
$592$	4052.52	0.281347
$593$	−3590.59	−0.248648	−0.124324	−	0.992242i	$-0.539676\pi$
$593$	−3590.59	−0.248648	−0.124324	+	0.992242i	$0.539676\pi$
$594$	0	0
$595$	1620.01	0.111620
$596$	−8265.06	−0.568037
$597$	0	0
$598$	−355.668	−0.0243217
$599$	−13703.5	−0.934743	−0.467372	−	0.884061i	$-0.654799\pi$
$599$	−13703.5	−0.934743	−0.467372	+	0.884061i	$0.654799\pi$
$600$	0	0
$601$	22807.7	1.54800	0.773999	−	0.633186i	$-0.218253\pi$
$601$	22807.7	1.54800	0.773999	+	0.633186i	$0.218253\pi$
$602$	−9418.75	−0.637674
$603$	0	0
$604$	4566.01	0.307596
$605$	76.8689	0.00516557
$606$	0	0
$607$	15378.7	1.02834	0.514171	−	0.857688i	$-0.328100\pi$
$607$	15378.7	1.02834	0.514171	+	0.857688i	$0.328100\pi$
$608$	−11891.1	−0.793170
$609$	0	0
$610$	381.291	0.0253082
$611$	−273.428	−0.0181043
$612$	0	0
$613$	7798.42	0.513826	0.256913	−	0.966435i	$-0.417295\pi$
$613$	7798.42	0.513826	0.256913	+	0.966435i	$0.417295\pi$
$614$	5816.56	0.382309
$615$	0	0
$616$	−29104.8	−1.90368
$617$	395.194	0.0257859	0.0128930	−	0.999917i	$-0.495896\pi$
$617$	395.194	0.0257859	0.0128930	+	0.999917i	$0.495896\pi$
$618$	0	0
$619$	−19278.1	−1.25178	−0.625890	−	0.779911i	$-0.715264\pi$
$619$	−19278.1	−1.25178	−0.625890	+	0.779911i	$0.715264\pi$
$620$	−86.3402	−0.00559275
$621$	0	0
$622$	−18372.5	−1.18435
$623$	−9436.28	−0.606833
$624$	0	0
$625$	15427.0	0.987325
$626$	−1895.34	−0.121011
$627$	0	0
$628$	4130.09	0.262434
$629$	−14061.9	−0.891389
$630$	0	0
$631$	−9549.87	−0.602495	−0.301247	−	0.953546i	$-0.597403\pi$
$631$	−9549.87	−0.602495	−0.301247	+	0.953546i	$0.597403\pi$
$632$	−23619.9	−1.48663
$633$	0	0
$634$	4119.11	0.258030
$635$	242.224	0.0151376
$636$	0	0
$637$	4232.09	0.263237
$638$	−16991.9	−1.05442
$639$	0	0
$640$	168.599	0.0104132
$641$	25966.8	1.60004	0.800022	−	0.599971i	$-0.204821\pi$
$641$	25966.8	1.60004	0.800022	+	0.599971i	$0.204821\pi$
$642$	0	0
$643$	11097.3	0.680616	0.340308	−	0.940314i	$-0.389469\pi$
$643$	11097.3	0.680616	0.340308	+	0.940314i	$0.389469\pi$
$644$	−3172.41	−0.194116
$645$	0	0
$646$	−11366.9	−0.692298
$647$	25.6058	0.00155590	0.000777951	−	1.00000i	$-0.499752\pi$
$647$	25.6058	0.00155590	0.000777951		1.00000i	$0.499752\pi$
$648$	0	0
$649$	30772.9	1.86124
$650$	1648.33	0.0994658
$651$	0	0
$652$	11304.0	0.678984
$653$	−18085.2	−1.08381	−0.541904	−	0.840440i	$-0.682297\pi$
$653$	−18085.2	−1.08381	−0.541904	+	0.840440i	$0.682297\pi$
$654$	0	0
$655$	1696.90	0.101227
$656$	8312.91	0.494763
$657$	0	0
$658$	2793.66	0.165514
$659$	13479.3	0.796779	0.398390	−	0.917216i	$-0.369569\pi$
$659$	13479.3	0.796779	0.398390	+	0.917216i	$0.369569\pi$
$660$	0	0
$661$	−18158.8	−1.06852	−0.534261	−	0.845319i	$-0.679410\pi$
$661$	−18158.8	−1.06852	−0.534261	+	0.845319i	$0.679410\pi$
$662$	23620.1	1.38674
$663$	0	0
$664$	−4903.10	−0.286562
$665$	1801.81	0.105070
$666$	0	0
$667$	−5825.77	−0.338193
$668$	−4302.50	−0.249205
$669$	0	0
$670$	−133.824	−0.00771650
$671$	9615.81	0.553225
$672$	0	0
$673$	−17475.3	−1.00093	−0.500463	−	0.865758i	$-0.666837\pi$
$673$	−17475.3	−1.00093	−0.500463	+	0.865758i	$0.666837\pi$
$674$	16640.8	0.951011
$675$	0	0
$676$	8039.03	0.457387
$677$	−21072.1	−1.19626	−0.598130	−	0.801399i	$-0.704089\pi$
$677$	−21072.1	−1.19626	−0.598130	+	0.801399i	$0.704089\pi$
$678$	0	0
$679$	80.9433	0.00457485
$680$	1239.63	0.0699084
$681$	0	0
$682$	2494.18	0.140040
$683$	−3477.49	−0.194821	−0.0974103	−	0.995244i	$-0.531056\pi$
$683$	−3477.49	−0.194821	−0.0974103	+	0.995244i	$0.531056\pi$
$684$	0	0
$685$	−242.399	−0.0135205
$686$	−20784.4	−1.15678
$687$	0	0
$688$	−2915.63	−0.161566
$689$	1323.53	0.0731821
$690$	0	0
$691$	−2554.94	−0.140658	−0.0703289	−	0.997524i	$-0.522405\pi$
$691$	−2554.94	−0.140658	−0.0703289	+	0.997524i	$0.522405\pi$
$692$	3428.33	0.188332
$693$	0	0
$694$	−1316.31	−0.0719980
$695$	−862.434	−0.0470705
$696$	0	0
$697$	−28845.1	−1.56755
$698$	4993.13	0.270763
$699$	0	0
$700$	14702.4	0.793855
$701$	19601.6	1.05612	0.528061	−	0.849207i	$-0.322919\pi$
$701$	19601.6	1.05612	0.528061	+	0.849207i	$0.322919\pi$
$702$	0	0
$703$	−15640.0	−0.839079
$704$	−18055.0	−0.966582
$705$	0	0
$706$	19775.6	1.05420
$707$	−4233.82	−0.225218
$708$	0	0
$709$	31659.7	1.67702	0.838508	−	0.544890i	$-0.183429\pi$
$709$	31659.7	1.67702	0.838508	+	0.544890i	$0.183429\pi$
$710$	1044.30	0.0552001
$711$	0	0
$712$	−7220.66	−0.380064
$713$	855.143	0.0449163
$714$	0	0
$715$	−176.628	−0.00923848
$716$	−12510.2	−0.652973
$717$	0	0
$718$	16457.4	0.855410
$719$	28280.1	1.46686	0.733429	−	0.679766i	$-0.237918\pi$
$719$	28280.1	1.46686	0.733429	+	0.679766i	$0.237918\pi$
$720$	0	0
$721$	−1355.61	−0.0700214
$722$	1532.91	0.0790154
$723$	0	0
$724$	−2279.51	−0.117013
$725$	26999.3	1.38307
$726$	0	0
$727$	13464.2	0.686877	0.343438	−	0.939175i	$-0.388408\pi$
$727$	13464.2	0.686877	0.343438	+	0.939175i	$0.388408\pi$
$728$	4920.18	0.250486
$729$	0	0
$730$	−948.197	−0.0480744
$731$	10117.0	0.511887
$732$	0	0
$733$	7272.98	0.366485	0.183243	−	0.983068i	$-0.441341\pi$
$733$	7272.98	0.366485	0.183243	+	0.983068i	$0.441341\pi$
$734$	6773.95	0.340642
$735$	0	0
$736$	−4083.31	−0.204501
$737$	−3374.91	−0.168679
$738$	0	0
$739$	7103.37	0.353588	0.176794	−	0.984248i	$-0.443427\pi$
$739$	7103.37	0.353588	0.176794	+	0.984248i	$0.443427\pi$
$740$	542.251	0.0269372
$741$	0	0
$742$	−13522.7	−0.669049
$743$	−18061.3	−0.891799	−0.445899	−	0.895083i	$-0.647116\pi$
$743$	−18061.3	−0.891799	−0.445899	+	0.895083i	$0.647116\pi$
$744$	0	0
$745$	1611.97	0.0792725
$746$	−3616.73	−0.177504
$747$	0	0
$748$	9938.84	0.485829
$749$	6256.68	0.305226
$750$	0	0
$751$	−31552.1	−1.53309	−0.766546	−	0.642189i	$-0.778026\pi$
$751$	−31552.1	−1.53309	−0.766546	+	0.642189i	$0.778026\pi$
$752$	864.793	0.0419358
$753$	0	0
$754$	2872.49	0.138740
$755$	−890.529	−0.0429267
$756$	0	0
$757$	−33184.8	−1.59329	−0.796647	−	0.604445i	$-0.793395\pi$
$757$	−33184.8	−1.59329	−0.796647	+	0.604445i	$0.793395\pi$
$758$	21188.5	1.01530
$759$	0	0
$760$	1378.75	0.0658059
$761$	−16789.7	−0.799772	−0.399886	−	0.916565i	$-0.630950\pi$
$761$	−16789.7	−0.799772	−0.399886	+	0.916565i	$0.630950\pi$
$762$	0	0
$763$	41078.3	1.94906
$764$	−1223.22	−0.0579248
$765$	0	0
$766$	17330.3	0.817455
$767$	−5202.17	−0.244902
$768$	0	0
$769$	32368.0	1.51784	0.758921	−	0.651182i	$-0.225727\pi$
$769$	32368.0	1.51784	0.758921	+	0.651182i	$0.225727\pi$
$770$	1804.64	0.0844605
$771$	0	0
$772$	−19673.7	−0.917190
$773$	−10661.1	−0.496060	−0.248030	−	0.968752i	$-0.579783\pi$
$773$	−10661.1	−0.496060	−0.248030	+	0.968752i	$0.579783\pi$
$774$	0	0
$775$	−3963.12	−0.183690
$776$	61.9379	0.00286526
$777$	0	0
$778$	−31438.5	−1.44875
$779$	−32082.2	−1.47556
$780$	0	0
$781$	26336.4	1.20665
$782$	−3903.30	−0.178493
$783$	0	0
$784$	−13385.2	−0.609748
$785$	−805.509	−0.0366240
$786$	0	0
$787$	30624.6	1.38710	0.693552	−	0.720407i	$-0.256045\pi$
$787$	30624.6	1.38710	0.693552	+	0.720407i	$0.256045\pi$
$788$	−9834.13	−0.444577
$789$	0	0
$790$	1464.54	0.0659571
$791$	−4133.91	−0.185822
$792$	0	0
$793$	−1625.56	−0.0727934
$794$	20067.1	0.896919
$795$	0	0
$796$	−605.244	−0.0269501
$797$	19757.5	0.878100	0.439050	−	0.898463i	$-0.355315\pi$
$797$	19757.5	0.878100	0.439050	+	0.898463i	$0.355315\pi$
$798$	0	0
$799$	−3000.75	−0.132865
$800$	18923.9	0.836327
$801$	0	0
$802$	17432.7	0.767543
$803$	−23912.7	−1.05088
$804$	0	0
$805$	618.729	0.0270898
$806$	−421.643	−0.0184265
$807$	0	0
$808$	−3239.73	−0.141056
$809$	2225.85	0.0967327	0.0483664	−	0.998830i	$-0.484599\pi$
$809$	2225.85	0.0967327	0.0483664	+	0.998830i	$0.484599\pi$
$810$	0	0
$811$	41746.2	1.80753	0.903766	−	0.428027i	$-0.140791\pi$
$811$	41746.2	1.80753	0.903766	+	0.428027i	$0.140791\pi$
$812$	25621.4	1.10731
$813$	0	0
$814$	−15664.5	−0.674496
$815$	−2204.66	−0.0947558
$816$	0	0
$817$	11252.3	0.481848
$818$	15294.8	0.653754
$819$	0	0
$820$	1112.32	0.0473705
$821$	−5021.19	−0.213448	−0.106724	−	0.994289i	$-0.534036\pi$
$821$	−5021.19	−0.213448	−0.106724	+	0.994289i	$0.534036\pi$
$822$	0	0
$823$	−26537.0	−1.12396	−0.561982	−	0.827149i	$-0.689961\pi$
$823$	−26537.0	−1.12396	−0.561982	+	0.827149i	$0.689961\pi$
$824$	−1037.31	−0.0438550
$825$	0	0
$826$	53151.4	2.23895
$827$	−36238.7	−1.52375	−0.761876	−	0.647723i	$-0.775721\pi$
$827$	−36238.7	−1.52375	−0.761876	+	0.647723i	$0.775721\pi$
$828$	0	0
$829$	31556.2	1.32207	0.661034	−	0.750356i	$-0.270118\pi$
$829$	31556.2	1.32207	0.661034	+	0.750356i	$0.270118\pi$
$830$	304.016	0.0127139
$831$	0	0
$832$	3052.21	0.127183
$833$	46445.4	1.93186
$834$	0	0
$835$	839.135	0.0347778
$836$	11054.2	0.457319
$837$	0	0
$838$	−5431.51	−0.223900
$839$	33671.9	1.38556	0.692779	−	0.721150i	$-0.256386\pi$
$839$	33671.9	1.38556	0.692779	+	0.721150i	$0.256386\pi$
$840$	0	0
$841$	22661.9	0.929184
$842$	−17214.3	−0.704565
$843$	0	0
$844$	−2606.43	−0.106300
$845$	−1567.89	−0.0638307
$846$	0	0
$847$	3348.32	0.135832
$848$	−4186.04	−0.169515
$849$	0	0
$850$	18089.7	0.729966
$851$	−5370.64	−0.216338
$852$	0	0
$853$	−41095.5	−1.64957	−0.824784	−	0.565448i	$-0.808703\pi$
$853$	−41095.5	−1.64957	−0.824784	+	0.565448i	$0.808703\pi$
$854$	16608.6	0.665496
$855$	0	0
$856$	4787.62	0.191165
$857$	24419.7	0.973351	0.486675	−	0.873583i	$-0.338209\pi$
$857$	24419.7	0.973351	0.486675	+	0.873583i	$0.338209\pi$
$858$	0	0
$859$	36957.6	1.46796	0.733979	−	0.679172i	$-0.237661\pi$
$859$	36957.6	1.46796	0.733979	+	0.679172i	$0.237661\pi$
$860$	−390.129	−0.0154689
$861$	0	0
$862$	−6956.05	−0.274854
$863$	19229.7	0.758502	0.379251	−	0.925294i	$-0.376182\pi$
$863$	19229.7	0.758502	0.379251	+	0.925294i	$0.376182\pi$
$864$	0	0
$865$	−668.642	−0.0262827
$866$	−15125.4	−0.593514
$867$	0	0
$868$	−3760.87	−0.147065
$869$	36934.5	1.44179
$870$	0	0
$871$	570.530	0.0221948
$872$	31433.1	1.22071
$873$	0	0
$874$	−4341.35	−0.168019
$875$	−5747.13	−0.222044
$876$	0	0
$877$	−14400.7	−0.554478	−0.277239	−	0.960801i	$-0.589419\pi$
$877$	−14400.7	−0.554478	−0.277239	+	0.960801i	$0.589419\pi$
$878$	−4836.99	−0.185923
$879$	0	0
$880$	558.636	0.0213996
$881$	22503.6	0.860574	0.430287	−	0.902692i	$-0.358412\pi$
$881$	22503.6	0.860574	0.430287	+	0.902692i	$0.358412\pi$
$882$	0	0
$883$	27155.3	1.03494	0.517468	−	0.855702i	$-0.326875\pi$
$883$	27155.3	1.03494	0.517468	+	0.855702i	$0.326875\pi$
$884$	−1680.17	−0.0639254
$885$	0	0
$886$	−5532.49	−0.209783
$887$	−10538.3	−0.398921	−0.199460	−	0.979906i	$-0.563919\pi$
$887$	−10538.3	−0.398921	−0.199460	+	0.979906i	$0.563919\pi$
$888$	0	0
$889$	10551.0	0.398053
$890$	447.715	0.0168623
$891$	0	0
$892$	3698.31	0.138821
$893$	−3337.51	−0.125068
$894$	0	0
$895$	2439.92	0.0911259
$896$	7343.97	0.273822
$897$	0	0
$898$	22321.9	0.829500
$899$	−6906.42	−0.256220
$900$	0	0
$901$	14525.2	0.537074
$902$	−32132.5	−1.18614
$903$	0	0
$904$	−3163.27	−0.116381
$905$	444.582	0.0163297
$906$	0	0
$907$	−41606.5	−1.52318	−0.761589	−	0.648061i	$-0.775580\pi$
$907$	−41606.5	−1.52318	−0.761589	+	0.648061i	$0.775580\pi$
$908$	15956.2	0.583177
$909$	0	0
$910$	−305.075	−0.0111133
$911$	−35548.5	−1.29284	−0.646418	−	0.762984i	$-0.723734\pi$
$911$	−35548.5	−1.29284	−0.646418	+	0.762984i	$0.723734\pi$
$912$	0	0
$913$	7666.99	0.277919
$914$	21664.8	0.784036
$915$	0	0
$916$	5624.43	0.202878
$917$	73915.0	2.66182
$918$	0	0
$919$	−49580.8	−1.77967	−0.889837	−	0.456278i	$-0.849182\pi$
$919$	−49580.8	−1.77967	−0.889837	+	0.456278i	$0.849182\pi$
$920$	473.452	0.0169666
$921$	0	0
$922$	−10603.8	−0.378759
$923$	−4452.18	−0.158771
$924$	0	0
$925$	24890.0	0.884734
$926$	1591.92	0.0564945
$927$	0	0
$928$	32978.2	1.16655
$929$	31127.8	1.09932	0.549660	−	0.835388i	$-0.314757\pi$
$929$	31127.8	1.09932	0.549660	+	0.835388i	$0.314757\pi$
$930$	0	0
$931$	51657.7	1.81849
$932$	16803.3	0.590570
$933$	0	0
$934$	−26164.6	−0.916628
$935$	−1938.42	−0.0678000
$936$	0	0
$937$	38617.1	1.34639	0.673194	−	0.739466i	$-0.264922\pi$
$937$	38617.1	1.34639	0.673194	+	0.739466i	$0.264922\pi$
$938$	−5829.19	−0.202910
$939$	0	0
$940$	115.715	0.00401510
$941$	−4407.19	−0.152678	−0.0763391	−	0.997082i	$-0.524323\pi$
$941$	−4407.19	−0.152678	−0.0763391	+	0.997082i	$0.524323\pi$
$942$	0	0
$943$	−11016.8	−0.380441
$944$	16453.3	0.567277
$945$	0	0
$946$	11270.0	0.387335
$947$	−34467.4	−1.18272	−0.591362	−	0.806406i	$-0.701409\pi$
$947$	−34467.4	−1.18272	−0.591362	+	0.806406i	$0.701409\pi$
$948$	0	0
$949$	4042.44	0.138275
$950$	20119.8	0.687129
$951$	0	0
$952$	53996.9	1.83829
$953$	−25665.4	−0.872385	−0.436193	−	0.899853i	$-0.643673\pi$
$953$	−25665.4	−0.872385	−0.436193	+	0.899853i	$0.643673\pi$
$954$	0	0
$955$	238.570	0.00808371
$956$	891.178	0.0301493
$957$	0	0
$958$	7416.02	0.250105
$959$	−10558.6	−0.355531
$960$	0	0
$961$	−28777.2	−0.965971
$962$	2648.09	0.0887502
$963$	0	0
$964$	12002.8	0.401021
$965$	3837.04	0.127999
$966$	0	0
$967$	49205.3	1.63634	0.818168	−	0.574980i	$-0.194990\pi$
$967$	49205.3	1.63634	0.818168	+	0.574980i	$0.194990\pi$
$968$	2562.14	0.0850725
$969$	0	0
$970$	−3.84044	−0.000127123 0
$971$	−31581.5	−1.04377	−0.521885	−	0.853016i	$-0.674771\pi$
$971$	−31581.5	−1.04377	−0.521885	+	0.853016i	$0.674771\pi$
$972$	0	0
$973$	−37566.6	−1.23775
$974$	9155.65	0.301197
$975$	0	0
$976$	5141.28	0.168615
$977$	−22410.8	−0.733863	−0.366932	−	0.930248i	$-0.619592\pi$
$977$	−22410.8	−0.733863	−0.366932	+	0.930248i	$0.619592\pi$
$978$	0	0
$979$	11291.0	0.368601
$980$	−1791.02	−0.0583796
$981$	0	0
$982$	−4319.35	−0.140363
$983$	−581.867	−0.0188796	−0.00943982	−	0.999955i	$-0.503005\pi$
$983$	−581.867	−0.0188796	−0.00943982	+	0.999955i	$0.503005\pi$
$984$	0	0
$985$	1917.99	0.0620430
$986$	31524.4	1.01820
$987$	0	0
$988$	−1868.72	−0.0601741
$989$	3863.97	0.124234
$990$	0	0
$991$	−19943.4	−0.639276	−0.319638	−	0.947540i	$-0.603561\pi$
$991$	−19943.4	−0.639276	−0.319638	+	0.947540i	$0.603561\pi$
$992$	−4840.74	−0.154933
$993$	0	0
$994$	45488.6	1.45152
$995$	118.043	0.00376103
$996$	0	0
$997$	13684.2	0.434687	0.217344	−	0.976095i	$-0.430261\pi$
$997$	13684.2	0.434687	0.217344	+	0.976095i	$0.430261\pi$
$998$	12496.4	0.396359
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.13		32
3.2	odd	2		717.4.a.d.1.20	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.20	✓	32	3.2	odd	2
2151.4.a.d.1.13		32	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)

\(f(q)\)	\(=\)	\(q-2.06669 q^{2} -3.72878 q^{4} +0.727240 q^{5} +31.6777 q^{7} +24.2398 q^{8} +O(q^{10})\)	\(q-2.06669 q^{2} -3.72878 q^{4} +0.727240 q^{5} +31.6777 q^{7} +24.2398 q^{8} -1.50298 q^{10} -37.9038 q^{11} +6.40766 q^{13} -65.4680 q^{14} -20.2660 q^{16} +70.3212 q^{17} +78.2130 q^{19} -2.71171 q^{20} +78.3356 q^{22} +26.8578 q^{23} -124.471 q^{25} -13.2427 q^{26} -118.119 q^{28} -216.912 q^{29} +31.8397 q^{31} -152.035 q^{32} -145.332 q^{34} +23.0373 q^{35} -199.966 q^{37} -161.642 q^{38} +17.6281 q^{40} -410.190 q^{41} +143.868 q^{43} +141.335 q^{44} -55.5068 q^{46} -42.6721 q^{47} +660.475 q^{49} +257.244 q^{50} -23.8927 q^{52} +206.555 q^{53} -27.5652 q^{55} +767.860 q^{56} +448.291 q^{58} -811.868 q^{59} -253.690 q^{61} -65.8029 q^{62} +476.337 q^{64} +4.65990 q^{65} +89.0387 q^{67} -262.212 q^{68} -47.6110 q^{70} -694.822 q^{71} +630.877 q^{73} +413.269 q^{74} -291.639 q^{76} -1200.70 q^{77} -974.426 q^{79} -14.7382 q^{80} +847.737 q^{82} -202.275 q^{83} +51.1404 q^{85} -297.331 q^{86} -918.781 q^{88} -297.884 q^{89} +202.980 q^{91} -100.147 q^{92} +88.1901 q^{94} +56.8796 q^{95} +2.55522 q^{97} -1365.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

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