LMFDB - Embedded newform 2151.4.a.h.1.3

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:	$0$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-5.15423 q^{2} +18.5661 q^{4} +1.27174 q^{5} -23.0732 q^{7} -54.4599 q^{8} +O(q^{10})$	\(q-5.15423 q^{2} +18.5661 q^{4} +1.27174 q^{5} -23.0732 q^{7} -54.4599 q^{8} -6.55481 q^{10} -4.21412 q^{11} -16.3792 q^{13} +118.925 q^{14} +132.170 q^{16} -98.0605 q^{17} -60.2154 q^{19} +23.6111 q^{20} +21.7205 q^{22} +168.855 q^{23} -123.383 q^{25} +84.4224 q^{26} -428.378 q^{28} +72.3381 q^{29} -26.8941 q^{31} -245.556 q^{32} +505.426 q^{34} -29.3430 q^{35} -86.2049 q^{37} +310.364 q^{38} -69.2585 q^{40} -137.305 q^{41} -406.867 q^{43} -78.2396 q^{44} -870.316 q^{46} -382.766 q^{47} +189.373 q^{49} +635.942 q^{50} -304.098 q^{52} -173.620 q^{53} -5.35925 q^{55} +1256.56 q^{56} -372.847 q^{58} +425.847 q^{59} -508.759 q^{61} +138.618 q^{62} +208.290 q^{64} -20.8301 q^{65} -650.419 q^{67} -1820.60 q^{68} +151.241 q^{70} -132.426 q^{71} -291.054 q^{73} +444.320 q^{74} -1117.96 q^{76} +97.2333 q^{77} +948.306 q^{79} +168.085 q^{80} +707.699 q^{82} -585.449 q^{83} -124.707 q^{85} +2097.09 q^{86} +229.501 q^{88} -587.676 q^{89} +377.922 q^{91} +3134.97 q^{92} +1972.86 q^{94} -76.5781 q^{95} -479.173 q^{97} -976.069 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) $ 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8	$ 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8} - 36 q^{10} + 132 q^{11} + 104 q^{13} + 280 q^{14} + 822 q^{16} + 408 q^{17} + 20 q^{19} + 800 q^{20} - 2 q^{22} + 276 q^{23} + 1477 q^{25} + 780 q^{26} + 224 q^{28} + 696 q^{29} - 380 q^{31} + 896 q^{32} - 72 q^{34} + 700 q^{35} + 224 q^{37} + 988 q^{38} - 258 q^{40} + 2706 q^{41} - 156 q^{43} + 1584 q^{44} + 428 q^{46} + 1316 q^{47} + 2135 q^{49} + 1400 q^{50} + 1092 q^{52} + 1484 q^{53} - 992 q^{55} + 3360 q^{56} - 120 q^{58} + 3186 q^{59} - 254 q^{61} + 1240 q^{62} + 3054 q^{64} + 5120 q^{65} + 288 q^{67} + 9420 q^{68} + 1108 q^{70} + 4468 q^{71} - 1770 q^{73} + 6214 q^{74} + 720 q^{76} + 6352 q^{77} - 746 q^{79} + 7040 q^{80} + 276 q^{82} + 5484 q^{83} + 588 q^{85} + 10152 q^{86} + 1186 q^{88} + 11570 q^{89} + 1768 q^{91} + 15366 q^{92} - 2142 q^{94} + 5736 q^{95} + 2390 q^{97} + 6912 q^{98}+O(q^{100}) $ 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8 - 36 * q^10 + 132 * q^11 + 104 * q^13 + 280 * q^14 + 822 * q^16 + 408 * q^17 + 20 * q^19 + 800 * q^20 - 2 * q^22 + 276 * q^23 + 1477 * q^25 + 780 * q^26 + 224 * q^28 + 696 * q^29 - 380 * q^31 + 896 * q^32 - 72 * q^34 + 700 * q^35 + 224 * q^37 + 988 * q^38 - 258 * q^40 + 2706 * q^41 - 156 * q^43 + 1584 * q^44 + 428 * q^46 + 1316 * q^47 + 2135 * q^49 + 1400 * q^50 + 1092 * q^52 + 1484 * q^53 - 992 * q^55 + 3360 * q^56 - 120 * q^58 + 3186 * q^59 - 254 * q^61 + 1240 * q^62 + 3054 * q^64 + 5120 * q^65 + 288 * q^67 + 9420 * q^68 + 1108 * q^70 + 4468 * q^71 - 1770 * q^73 + 6214 * q^74 + 720 * q^76 + 6352 * q^77 - 746 * q^79 + 7040 * q^80 + 276 * q^82 + 5484 * q^83 + 588 * q^85 + 10152 * q^86 + 1186 * q^88 + 11570 * q^89 + 1768 * q^91 + 15366 * q^92 - 2142 * q^94 + 5736 * q^95 + 2390 * q^97 + 6912 * 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 $

$2$	−5.15423	−1.82229	−0.911147	−	0.412081i	$-0.864802\pi$
$2$	−5.15423	−1.82229	−0.911147	+	0.412081i	$0.864802\pi$
$3$	0	0
$3$	0	0
$4$	18.5661	2.32076
$5$	1.27174	0.113747	0.0568737	−	0.998381i	$-0.481887\pi$
$5$	1.27174	0.113747	0.0568737	+	0.998381i	$0.481887\pi$
$6$	0	0
$7$	−23.0732	−1.24584	−0.622918	−	0.782287i	$-0.714053\pi$
$7$	−23.0732	−1.24584	−0.622918	+	0.782287i	$0.714053\pi$
$8$	−54.4599	−2.40681
$9$	0	0
$10$	−6.55481	−0.207281
$11$	−4.21412	−0.115510	−0.0577548	−	0.998331i	$-0.518394\pi$
$11$	−4.21412	−0.115510	−0.0577548	+	0.998331i	$0.518394\pi$
$12$	0	0
$13$	−16.3792	−0.349445	−0.174723	−	0.984618i	$-0.555903\pi$
$13$	−16.3792	−0.349445	−0.174723	+	0.984618i	$0.555903\pi$
$14$	118.925	2.27028
$15$	0	0
$16$	132.170	2.06516
$17$	−98.0605	−1.39901	−0.699505	−	0.714628i	$-0.746596\pi$
$17$	−98.0605	−1.39901	−0.699505	+	0.714628i	$0.746596\pi$
$18$	0	0
$19$	−60.2154	−0.727072	−0.363536	−	0.931580i	$-0.618431\pi$
$19$	−60.2154	−0.727072	−0.363536	+	0.931580i	$0.618431\pi$
$20$	23.6111	0.263980
$21$	0	0
$22$	21.7205	0.210493
$23$	168.855	1.53081	0.765406	−	0.643548i	$-0.222538\pi$
$23$	168.855	1.53081	0.765406	+	0.643548i	$0.222538\pi$
$24$	0	0
$25$	−123.383	−0.987062
$26$	84.4224	0.636792
$27$	0	0
$28$	−428.378	−2.89128
$29$	72.3381	0.463202	0.231601	−	0.972811i	$-0.425604\pi$
$29$	72.3381	0.463202	0.231601	+	0.972811i	$0.425604\pi$
$30$	0	0
$31$	−26.8941	−0.155817	−0.0779084	−	0.996961i	$-0.524824\pi$
$31$	−26.8941	−0.155817	−0.0779084	+	0.996961i	$0.524824\pi$
$32$	−245.556	−1.35652
$33$	0	0
$34$	505.426	2.54941
$35$	−29.3430	−0.141711
$36$	0	0
$37$	−86.2049	−0.383027	−0.191514	−	0.981490i	$-0.561340\pi$
$37$	−86.2049	−0.383027	−0.191514	+	0.981490i	$0.561340\pi$
$38$	310.364	1.32494
$39$	0	0
$40$	−69.2585	−0.273768
$41$	−137.305	−0.523009	−0.261505	−	0.965202i	$-0.584219\pi$
$41$	−137.305	−0.523009	−0.261505	+	0.965202i	$0.584219\pi$
$42$	0	0
$43$	−406.867	−1.44295	−0.721473	−	0.692443i	$-0.756535\pi$
$43$	−406.867	−1.44295	−0.721473	+	0.692443i	$0.756535\pi$
$44$	−78.2396	−0.268070
$45$	0	0
$46$	−870.316	−2.78959
$47$	−382.766	−1.18792	−0.593959	−	0.804495i	$-0.702436\pi$
$47$	−382.766	−1.18792	−0.593959	+	0.804495i	$0.702436\pi$
$48$	0	0
$49$	189.373	0.552107
$50$	635.942	1.79872
$51$	0	0
$52$	−304.098	−0.810977
$53$	−173.620	−0.449972	−0.224986	−	0.974362i	$-0.572234\pi$
$53$	−173.620	−0.449972	−0.224986	+	0.974362i	$0.572234\pi$
$54$	0	0
$55$	−5.35925	−0.0131389
$56$	1256.56	2.99849
$57$	0	0
$58$	−372.847	−0.844090
$59$	425.847	0.939670	0.469835	−	0.882754i	$-0.344313\pi$
$59$	425.847	0.939670	0.469835	+	0.882754i	$0.344313\pi$
$60$	0	0
$61$	−508.759	−1.06787	−0.533933	−	0.845527i	$-0.679287\pi$
$61$	−508.759	−1.06787	−0.533933	+	0.845527i	$0.679287\pi$
$62$	138.618	0.283944
$63$	0	0
$64$	208.290	0.406816
$65$	−20.8301	−0.0397485
$66$	0	0
$67$	−650.419	−1.18599	−0.592995	−	0.805206i	$-0.702055\pi$
$67$	−650.419	−1.18599	−0.592995	+	0.805206i	$0.702055\pi$
$68$	−1820.60	−3.24676
$69$	0	0
$70$	151.241	0.258239
$71$	−132.426	−0.221354	−0.110677	−	0.993856i	$-0.535302\pi$
$71$	−132.426	−0.221354	−0.110677	+	0.993856i	$0.535302\pi$
$72$	0	0
$73$	−291.054	−0.466648	−0.233324	−	0.972399i	$-0.574960\pi$
$73$	−291.054	−0.466648	−0.233324	+	0.972399i	$0.574960\pi$
$74$	444.320	0.697988
$75$	0	0
$76$	−1117.96	−1.68736
$77$	97.2333	0.143906
$78$	0	0
$79$	948.306	1.35054	0.675271	−	0.737570i	$-0.264027\pi$
$79$	948.306	1.35054	0.675271	+	0.737570i	$0.264027\pi$
$80$	168.085	0.234906
$81$	0	0
$82$	707.699	0.953077
$83$	−585.449	−0.774234	−0.387117	−	0.922031i	$-0.626529\pi$
$83$	−585.449	−0.774234	−0.387117	+	0.922031i	$0.626529\pi$
$84$	0	0
$85$	−124.707	−0.159134
$86$	2097.09	2.62947
$87$	0	0
$88$	229.501	0.278010
$89$	−587.676	−0.699928	−0.349964	−	0.936763i	$-0.613806\pi$
$89$	−587.676	−0.699928	−0.349964	+	0.936763i	$0.613806\pi$
$90$	0	0
$91$	377.922	0.435351
$92$	3134.97	3.55264
$93$	0	0
$94$	1972.86	2.16474
$95$	−76.5781	−0.0827026
$96$	0	0
$97$	−479.173	−0.501574	−0.250787	−	0.968042i	$-0.580689\pi$
$97$	−479.173	−0.501574	−0.250787	+	0.968042i	$0.580689\pi$
$98$	−976.069	−1.00610
$99$	0	0
$100$	−2290.73	−2.29073
$101$	1714.66	1.68926	0.844630	−	0.535350i	$-0.179820\pi$
$101$	1714.66	1.68926	0.844630	+	0.535350i	$0.179820\pi$
$102$	0	0
$103$	41.4624	0.0396642	0.0198321	−	0.999803i	$-0.493687\pi$
$103$	41.4624	0.0396642	0.0198321	+	0.999803i	$0.493687\pi$
$104$	892.012	0.841048
$105$	0	0
$106$	894.876	0.819982
$107$	−963.414	−0.870437	−0.435219	−	0.900325i	$-0.643329\pi$
$107$	−963.414	−0.870437	−0.435219	+	0.900325i	$0.643329\pi$
$108$	0	0
$109$	477.755	0.419822	0.209911	−	0.977720i	$-0.432683\pi$
$109$	477.755	0.419822	0.209911	+	0.977720i	$0.432683\pi$
$110$	27.6228	0.0239430
$111$	0	0
$112$	−3049.59	−2.57285
$113$	785.408	0.653850	0.326925	−	0.945050i	$-0.393988\pi$
$113$	785.408	0.653850	0.326925	+	0.945050i	$0.393988\pi$
$114$	0	0
$115$	214.739	0.174126
$116$	1343.03	1.07498
$117$	0	0
$118$	−2194.91	−1.71236
$119$	2262.57	1.74294
$120$	0	0
$121$	−1313.24	−0.986658
$122$	2622.26	1.94597
$123$	0	0
$124$	−499.317	−0.361613
$125$	−315.877	−0.226023
$126$	0	0
$127$	−847.836	−0.592388	−0.296194	−	0.955128i	$-0.595717\pi$
$127$	−847.836	−0.592388	−0.296194	+	0.955128i	$0.595717\pi$
$128$	890.874	0.615179
$129$	0	0
$130$	107.363	0.0724334
$131$	−390.738	−0.260602	−0.130301	−	0.991474i	$-0.541594\pi$
$131$	−390.738	−0.260602	−0.130301	+	0.991474i	$0.541594\pi$
$132$	0	0
$133$	1389.36	0.905812
$134$	3352.41	2.16122
$135$	0	0
$136$	5340.36	3.36715
$137$	2266.41	1.41338	0.706688	−	0.707525i	$-0.250188\pi$
$137$	2266.41	1.41338	0.706688	+	0.707525i	$0.250188\pi$
$138$	0	0
$139$	−1891.28	−1.15407	−0.577036	−	0.816719i	$-0.695791\pi$
$139$	−1891.28	−1.15407	−0.577036	+	0.816719i	$0.695791\pi$
$140$	−544.784	−0.328876
$141$	0	0
$142$	682.556	0.403372
$143$	69.0241	0.0403643
$144$	0	0
$145$	91.9949	0.0526880
$146$	1500.16	0.850369
$147$	0	0
$148$	−1600.49	−0.888913
$149$	790.756	0.434773	0.217387	−	0.976086i	$-0.430247\pi$
$149$	790.756	0.434773	0.217387	+	0.976086i	$0.430247\pi$
$150$	0	0
$151$	2688.63	1.44899	0.724495	−	0.689280i	$-0.242073\pi$
$151$	2688.63	1.44899	0.724495	+	0.689280i	$0.242073\pi$
$152$	3279.33	1.74992
$153$	0	0
$154$	−501.162	−0.262239
$155$	−34.2021	−0.0177238
$156$	0	0
$157$	−299.027	−0.152006	−0.0760031	−	0.997108i	$-0.524216\pi$
$157$	−299.027	−0.152006	−0.0760031	+	0.997108i	$0.524216\pi$
$158$	−4887.79	−2.46108
$159$	0	0
$160$	−312.282	−0.154300
$161$	−3896.02	−1.90714
$162$	0	0
$163$	148.627	0.0714195	0.0357098	−	0.999362i	$-0.488631\pi$
$163$	148.627	0.0714195	0.0357098	+	0.999362i	$0.488631\pi$
$164$	−2549.21	−1.21378
$165$	0	0
$166$	3017.54	1.41088
$167$	3277.62	1.51874	0.759372	−	0.650657i	$-0.225507\pi$
$167$	3277.62	1.51874	0.759372	+	0.650657i	$0.225507\pi$
$168$	0	0
$169$	−1928.72	−0.877888
$170$	642.768	0.289989
$171$	0	0
$172$	−7553.92	−3.34873
$173$	−3263.65	−1.43428	−0.717141	−	0.696928i	$-0.754550\pi$
$173$	−3263.65	−1.43428	−0.717141	+	0.696928i	$0.754550\pi$
$174$	0	0
$175$	2846.83	1.22972
$176$	−556.981	−0.238546
$177$	0	0
$178$	3029.02	1.27547
$179$	−4519.98	−1.88737	−0.943684	−	0.330848i	$-0.892665\pi$
$179$	−4519.98	−1.88737	−0.943684	+	0.330848i	$0.892665\pi$
$180$	0	0
$181$	−3600.29	−1.47849	−0.739247	−	0.673435i	$-0.764818\pi$
$181$	−3600.29	−1.47849	−0.739247	+	0.673435i	$0.764818\pi$
$182$	−1947.89	−0.793338
$183$	0	0
$184$	−9195.81	−3.68437
$185$	−109.630	−0.0435684
$186$	0	0
$187$	413.239	0.161599
$188$	−7106.46	−2.75687
$189$	0	0
$190$	394.701	0.150709
$191$	−1601.68	−0.606773	−0.303387	−	0.952868i	$-0.598117\pi$
$191$	−1601.68	−0.606773	−0.303387	+	0.952868i	$0.598117\pi$
$192$	0	0
$193$	−3459.52	−1.29027	−0.645134	−	0.764069i	$-0.723198\pi$
$193$	−3459.52	−1.29027	−0.645134	+	0.764069i	$0.723198\pi$
$194$	2469.77	0.914015
$195$	0	0
$196$	3515.90	1.28131
$197$	835.346	0.302111	0.151056	−	0.988525i	$-0.451733\pi$
$197$	835.346	0.302111	0.151056	+	0.988525i	$0.451733\pi$
$198$	0	0
$199$	−1250.75	−0.445545	−0.222772	−	0.974871i	$-0.571511\pi$
$199$	−1250.75	−0.445545	−0.222772	+	0.974871i	$0.571511\pi$
$200$	6719.41	2.37567
$201$	0	0
$202$	−8837.76	−3.07833
$203$	−1669.07	−0.577073
$204$	0	0
$205$	−174.615	−0.0594910
$206$	−213.707	−0.0722798
$207$	0	0
$208$	−2164.85	−0.721659
$209$	253.755	0.0839838
$210$	0	0
$211$	937.142	0.305761	0.152880	−	0.988245i	$-0.451145\pi$
$211$	937.142	0.305761	0.152880	+	0.988245i	$0.451145\pi$
$212$	−3223.44	−1.04428
$213$	0	0
$214$	4965.66	1.58619
$215$	−517.427	−0.164131
$216$	0	0
$217$	620.532	0.194122
$218$	−2462.46	−0.765039
$219$	0	0
$220$	−99.5001	−0.0304923
$221$	1606.16	0.488877
$222$	0	0
$223$	2589.67	0.777655	0.388827	−	0.921311i	$-0.372880\pi$
$223$	2589.67	0.777655	0.388827	+	0.921311i	$0.372880\pi$
$224$	5665.76	1.69000
$225$	0	0
$226$	−4048.17	−1.19151
$227$	−5121.76	−1.49755	−0.748773	−	0.662826i	$-0.769357\pi$
$227$	−5121.76	−1.49755	−0.748773	+	0.662826i	$0.769357\pi$
$228$	0	0
$229$	3590.34	1.03605	0.518027	−	0.855364i	$-0.326666\pi$
$229$	3590.34	1.03605	0.518027	+	0.855364i	$0.326666\pi$
$230$	−1106.81	−0.317309
$231$	0	0
$232$	−3939.52	−1.11484
$233$	−883.730	−0.248477	−0.124238	−	0.992252i	$-0.539649\pi$
$233$	−883.730	−0.248477	−0.124238	+	0.992252i	$0.539649\pi$
$234$	0	0
$235$	−486.777	−0.135123
$236$	7906.30	2.18075
$237$	0	0
$238$	−11661.8	−3.17614
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−1447.09	−0.386784	−0.193392	−	0.981122i	$-0.561949\pi$
$241$	−1447.09	−0.386784	−0.193392	+	0.981122i	$0.561949\pi$
$242$	6768.74	1.79798
$243$	0	0
$244$	−9445.64	−2.47826
$245$	240.832	0.0628007
$246$	0	0
$247$	986.284	0.254072
$248$	1464.65	0.375021
$249$	0	0
$250$	1628.10	0.411881
$251$	743.264	0.186910	0.0934550	−	0.995624i	$-0.470209\pi$
$251$	743.264	0.186910	0.0934550	+	0.995624i	$0.470209\pi$
$252$	0	0
$253$	−711.575	−0.176823
$254$	4369.94	1.07951
$255$	0	0
$256$	−6258.09	−1.52785
$257$	2369.85	0.575204	0.287602	−	0.957750i	$-0.407142\pi$
$257$	2369.85	0.575204	0.287602	+	0.957750i	$0.407142\pi$
$258$	0	0
$259$	1989.02	0.477189
$260$	−386.732	−0.0922466
$261$	0	0
$262$	2013.95	0.474894
$263$	−6429.45	−1.50744	−0.753720	−	0.657196i	$-0.771742\pi$
$263$	−6429.45	−1.50744	−0.753720	+	0.657196i	$0.771742\pi$
$264$	0	0
$265$	−220.799	−0.0511832
$266$	−7161.09	−1.65066
$267$	0	0
$268$	−12075.7	−2.75240
$269$	1740.36	0.394468	0.197234	−	0.980356i	$-0.436804\pi$
$269$	1740.36	0.394468	0.197234	+	0.980356i	$0.436804\pi$
$270$	0	0
$271$	965.726	0.216471	0.108236	−	0.994125i	$-0.465480\pi$
$271$	965.726	0.216471	0.108236	+	0.994125i	$0.465480\pi$
$272$	−12960.7	−2.88918
$273$	0	0
$274$	−11681.6	−2.57559
$275$	519.950	0.114015
$276$	0	0
$277$	−1837.77	−0.398631	−0.199315	−	0.979935i	$-0.563872\pi$
$277$	−1837.77	−0.398631	−0.199315	+	0.979935i	$0.563872\pi$
$278$	9748.07	2.10306
$279$	0	0
$280$	1598.02	0.341071
$281$	−7292.56	−1.54818	−0.774088	−	0.633078i	$-0.781791\pi$
$281$	−7292.56	−1.54818	−0.774088	+	0.633078i	$0.781791\pi$
$282$	0	0
$283$	4293.55	0.901855	0.450927	−	0.892561i	$-0.351093\pi$
$283$	4293.55	0.901855	0.450927	+	0.892561i	$0.351093\pi$
$284$	−2458.64	−0.513709
$285$	0	0
$286$	−355.766	−0.0735556
$287$	3168.06	0.651583
$288$	0	0
$289$	4702.86	0.957228
$290$	−474.163	−0.0960131
$291$	0	0
$292$	−5403.72	−1.08298
$293$	−3253.10	−0.648629	−0.324315	−	0.945949i	$-0.605134\pi$
$293$	−3253.10	−0.648629	−0.324315	+	0.945949i	$0.605134\pi$
$294$	0	0
$295$	541.564	0.106885
$296$	4694.71	0.921873
$297$	0	0
$298$	−4075.74	−0.792285
$299$	−2765.71	−0.534934
$300$	0	0
$301$	9387.73	1.79767
$302$	−13857.8	−2.64049
$303$	0	0
$304$	−7958.68	−1.50152
$305$	−647.006	−0.121467
$306$	0	0
$307$	3914.79	0.727781	0.363890	−	0.931442i	$-0.381448\pi$
$307$	3914.79	0.727781	0.363890	+	0.931442i	$0.381448\pi$
$308$	1805.24	0.333971
$309$	0	0
$310$	176.286	0.0322979
$311$	3124.86	0.569757	0.284878	−	0.958564i	$-0.408047\pi$
$311$	3124.86	0.569757	0.284878	+	0.958564i	$0.408047\pi$
$312$	0	0
$313$	7914.98	1.42933	0.714667	−	0.699465i	$-0.246578\pi$
$313$	7914.98	1.42933	0.714667	+	0.699465i	$0.246578\pi$
$314$	1541.25	0.277000
$315$	0	0
$316$	17606.3	3.13428
$317$	−3512.36	−0.622314	−0.311157	−	0.950358i	$-0.600717\pi$
$317$	−3512.36	−0.622314	−0.311157	+	0.950358i	$0.600717\pi$
$318$	0	0
$319$	−304.842	−0.0535042
$320$	264.889	0.0462743
$321$	0	0
$322$	20081.0	3.47537
$323$	5904.76	1.01718
$324$	0	0
$325$	2020.92	0.344924
$326$	−766.058	−0.130147
$327$	0	0
$328$	7477.59	1.25878
$329$	8831.64	1.47995
$330$	0	0
$331$	8333.51	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$331$	8333.51	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$332$	−10869.5	−1.79681
$333$	0	0
$334$	−16893.6	−2.76760
$335$	−827.161	−0.134903
$336$	0	0
$337$	4668.20	0.754578	0.377289	−	0.926096i	$-0.376856\pi$
$337$	4668.20	0.754578	0.377289	+	0.926096i	$0.376856\pi$
$338$	9941.06	1.59977
$339$	0	0
$340$	−2315.32	−0.369311
$341$	113.335	0.0179983
$342$	0	0
$343$	3544.68	0.558002
$344$	22157.9	3.47290
$345$	0	0
$346$	16821.6	2.61369
$347$	−4398.94	−0.680541	−0.340270	−	0.940328i	$-0.610519\pi$
$347$	−4398.94	−0.680541	−0.340270	+	0.940328i	$0.610519\pi$
$348$	0	0
$349$	5610.61	0.860541	0.430271	−	0.902700i	$-0.358418\pi$
$349$	5610.61	0.860541	0.430271	+	0.902700i	$0.358418\pi$
$350$	−14673.2	−2.24091
$351$	0	0
$352$	1034.80	0.156691
$353$	97.4593	0.0146947	0.00734736	−	0.999973i	$-0.497661\pi$
$353$	97.4593	0.0146947	0.00734736	+	0.999973i	$0.497661\pi$
$354$	0	0
$355$	−168.411	−0.0251785
$356$	−10910.8	−1.62436
$357$	0	0
$358$	23297.0	3.43934
$359$	8080.31	1.18792	0.593958	−	0.804496i	$-0.297564\pi$
$359$	8080.31	1.18792	0.593958	+	0.804496i	$0.297564\pi$
$360$	0	0
$361$	−3233.10	−0.471366
$362$	18556.7	2.69425
$363$	0	0
$364$	7016.52	1.01034
$365$	−370.143	−0.0530800
$366$	0	0
$367$	4871.70	0.692918	0.346459	−	0.938065i	$-0.387384\pi$
$367$	4871.70	0.692918	0.346459	+	0.938065i	$0.387384\pi$
$368$	22317.6	3.16137
$369$	0	0
$370$	565.057	0.0793944
$371$	4005.97	0.560591
$372$	0	0
$373$	−7045.04	−0.977957	−0.488979	−	0.872296i	$-0.662630\pi$
$373$	−7045.04	−0.977957	−0.488979	+	0.872296i	$0.662630\pi$
$374$	−2129.93	−0.294481
$375$	0	0
$376$	20845.4	2.85909
$377$	−1184.84	−0.161864
$378$	0	0
$379$	5969.39	0.809043	0.404521	−	0.914529i	$-0.367438\pi$
$379$	5969.39	0.809043	0.404521	+	0.914529i	$0.367438\pi$
$380$	−1421.75	−0.191933
$381$	0	0
$382$	8255.44	1.10572
$383$	3617.38	0.482610	0.241305	−	0.970449i	$-0.422425\pi$
$383$	3617.38	0.482610	0.241305	+	0.970449i	$0.422425\pi$
$384$	0	0
$385$	123.655	0.0163689
$386$	17831.2	2.35125
$387$	0	0
$388$	−8896.36	−1.16403
$389$	3143.72	0.409750	0.204875	−	0.978788i	$-0.434321\pi$
$389$	3143.72	0.409750	0.204875	+	0.978788i	$0.434321\pi$
$390$	0	0
$391$	−16558.0	−2.14162
$392$	−10313.2	−1.32882
$393$	0	0
$394$	−4305.57	−0.550536
$395$	1205.99	0.153621
$396$	0	0
$397$	3961.64	0.500828	0.250414	−	0.968139i	$-0.419433\pi$
$397$	3961.64	0.500828	0.250414	+	0.968139i	$0.419433\pi$
$398$	6446.65	0.811913
$399$	0	0
$400$	−16307.5	−2.03844
$401$	5164.39	0.643135	0.321568	−	0.946887i	$-0.395790\pi$
$401$	5164.39	0.643135	0.321568	+	0.946887i	$0.395790\pi$
$402$	0	0
$403$	440.505	0.0544494
$404$	31834.5	3.92036
$405$	0	0
$406$	8602.77	1.05160
$407$	363.278	0.0442433
$408$	0	0
$409$	−15261.8	−1.84510	−0.922550	−	0.385879i	$-0.873898\pi$
$409$	−15261.8	−1.84510	−0.922550	+	0.385879i	$0.873898\pi$
$410$	900.006	0.108410
$411$	0	0
$412$	769.793	0.0920510
$413$	−9825.65	−1.17067
$414$	0	0
$415$	−744.537	−0.0880671
$416$	4022.02	0.474028
$417$	0	0
$418$	−1307.91	−0.153043
$419$	8757.77	1.02111	0.510555	−	0.859845i	$-0.329440\pi$
$419$	8757.77	1.02111	0.510555	+	0.859845i	$0.329440\pi$
$420$	0	0
$421$	−2925.07	−0.338620	−0.169310	−	0.985563i	$-0.554154\pi$
$421$	−2925.07	−0.338620	−0.169310	+	0.985563i	$0.554154\pi$
$422$	−4830.24	−0.557186
$423$	0	0
$424$	9455.32	1.08300
$425$	12099.0	1.38091
$426$	0	0
$427$	11738.7	1.33039
$428$	−17886.8	−2.02007
$429$	0	0
$430$	2666.94	0.299096
$431$	−5246.34	−0.586327	−0.293164	−	0.956062i	$-0.594708\pi$
$431$	−5246.34	−0.586327	−0.293164	+	0.956062i	$0.594708\pi$
$432$	0	0
$433$	−2901.54	−0.322030	−0.161015	−	0.986952i	$-0.551477\pi$
$433$	−2901.54	−0.322030	−0.161015	+	0.986952i	$0.551477\pi$
$434$	−3198.36	−0.353748
$435$	0	0
$436$	8870.02	0.974305
$437$	−10167.7	−1.11301
$438$	0	0
$439$	4567.53	0.496575	0.248287	−	0.968686i	$-0.420132\pi$
$439$	4567.53	0.496575	0.248287	+	0.968686i	$0.420132\pi$
$440$	291.864	0.0316229
$441$	0	0
$442$	−8278.50	−0.890878
$443$	−2082.15	−0.223309	−0.111654	−	0.993747i	$-0.535615\pi$
$443$	−2082.15	−0.223309	−0.111654	+	0.993747i	$0.535615\pi$
$444$	0	0
$445$	−747.369	−0.0796150
$446$	−13347.7	−1.41712
$447$	0	0
$448$	−4805.91	−0.506826
$449$	−64.4340	−0.00677245	−0.00338622	−	0.999994i	$-0.501078\pi$
$449$	−64.4340	−0.00677245	−0.00338622	+	0.999994i	$0.501078\pi$
$450$	0	0
$451$	578.618	0.0604126
$452$	14581.9	1.51743
$453$	0	0
$454$	26398.7	2.72897
$455$	480.616	0.0495201
$456$	0	0
$457$	−8606.13	−0.880915	−0.440457	−	0.897774i	$-0.645184\pi$
$457$	−8606.13	−0.880915	−0.440457	+	0.897774i	$0.645184\pi$
$458$	−18505.4	−1.88800
$459$	0	0
$460$	3986.85	0.404104
$461$	15415.3	1.55740	0.778698	−	0.627398i	$-0.215880\pi$
$461$	15415.3	1.55740	0.778698	+	0.627398i	$0.215880\pi$
$462$	0	0
$463$	10632.2	1.06722	0.533609	−	0.845732i	$-0.320836\pi$
$463$	10632.2	1.06722	0.533609	+	0.845732i	$0.320836\pi$
$464$	9560.94	0.956585
$465$	0	0
$466$	4554.95	0.452798
$467$	−2712.98	−0.268826	−0.134413	−	0.990925i	$-0.542915\pi$
$467$	−2712.98	−0.268826	−0.134413	+	0.990925i	$0.542915\pi$
$468$	0	0
$469$	15007.2	1.47755
$470$	2508.96	0.246233
$471$	0	0
$472$	−23191.6	−2.26161
$473$	1714.59	0.166674
$474$	0	0
$475$	7429.54	0.717665
$476$	42007.0	4.04493
$477$	0	0
$478$	1231.86	0.117874
$479$	9677.65	0.923138	0.461569	−	0.887104i	$-0.347287\pi$
$479$	9677.65	0.923138	0.461569	+	0.887104i	$0.347287\pi$
$480$	0	0
$481$	1411.97	0.133847
$482$	7458.61	0.704835
$483$	0	0
$484$	−24381.7	−2.28979
$485$	−609.381	−0.0570527
$486$	0	0
$487$	−1458.67	−0.135726	−0.0678630	−	0.997695i	$-0.521618\pi$
$487$	−1458.67	−0.135726	−0.0678630	+	0.997695i	$0.521618\pi$
$488$	27706.9	2.57015
$489$	0	0
$490$	−1241.30	−0.114441
$491$	−10497.8	−0.964884	−0.482442	−	0.875928i	$-0.660250\pi$
$491$	−10497.8	−0.964884	−0.482442	+	0.875928i	$0.660250\pi$
$492$	0	0
$493$	−7093.51	−0.648024
$494$	−5083.53	−0.462994
$495$	0	0
$496$	−3554.59	−0.321786
$497$	3055.50	0.275771
$498$	0	0
$499$	18915.4	1.69694	0.848468	−	0.529247i	$-0.177525\pi$
$499$	18915.4	1.69694	0.848468	+	0.529247i	$0.177525\pi$
$500$	−5864.59	−0.524545
$501$	0	0
$502$	−3830.95	−0.340605
$503$	−910.675	−0.0807257	−0.0403628	−	0.999185i	$-0.512851\pi$
$503$	−910.675	−0.0807257	−0.0403628	+	0.999185i	$0.512851\pi$
$504$	0	0
$505$	2180.60	0.192149
$506$	3667.62	0.322224
$507$	0	0
$508$	−15741.0	−1.37479
$509$	2067.73	0.180060	0.0900301	−	0.995939i	$-0.471304\pi$
$509$	2067.73	0.180060	0.0900301	+	0.995939i	$0.471304\pi$
$510$	0	0
$511$	6715.54	0.581366
$512$	25128.6	2.16902
$513$	0	0
$514$	−12214.8	−1.04819
$515$	52.7292	0.00451170
$516$	0	0
$517$	1613.02	0.137216
$518$	−10251.9	−0.869579
$519$	0	0
$520$	1134.40	0.0956670
$521$	9551.18	0.803157	0.401579	−	0.915825i	$-0.368462\pi$
$521$	9551.18	0.803157	0.401579	+	0.915825i	$0.368462\pi$
$522$	0	0
$523$	−22594.0	−1.88904	−0.944520	−	0.328455i	$-0.893472\pi$
$523$	−22594.0	−1.88904	−0.944520	+	0.328455i	$0.893472\pi$
$524$	−7254.46	−0.604795
$525$	0	0
$526$	33138.8	2.74700
$527$	2637.25	0.217989
$528$	0	0
$529$	16345.0	1.34338
$530$	1138.05	0.0932709
$531$	0	0
$532$	25795.0	2.10217
$533$	2248.95	0.182763
$534$	0	0
$535$	−1225.21	−0.0990100
$536$	35421.7	2.85445
$537$	0	0
$538$	−8970.23	−0.718837
$539$	−798.039	−0.0637736
$540$	0	0
$541$	10689.3	0.849479	0.424740	−	0.905316i	$-0.360366\pi$
$541$	10689.3	0.849479	0.424740	+	0.905316i	$0.360366\pi$
$542$	−4977.57	−0.394474
$543$	0	0
$544$	24079.3	1.89778
$545$	607.577	0.0477537
$546$	0	0
$547$	−18533.4	−1.44869	−0.724344	−	0.689439i	$-0.757857\pi$
$547$	−18533.4	−1.44869	−0.724344	+	0.689439i	$0.757857\pi$
$548$	42078.3	3.28010
$549$	0	0
$550$	−2679.94	−0.207769
$551$	−4355.87	−0.336781
$552$	0	0
$553$	−21880.5	−1.68255
$554$	9472.27	0.726423
$555$	0	0
$556$	−35113.5	−2.67832
$557$	18530.5	1.40963	0.704814	−	0.709393i	$-0.251031\pi$
$557$	18530.5	1.40963	0.704814	+	0.709393i	$0.251031\pi$
$558$	0	0
$559$	6664.18	0.504230
$560$	−3878.27	−0.292655
$561$	0	0
$562$	37587.5	2.82123
$563$	−12789.8	−0.957417	−0.478709	−	0.877974i	$-0.658895\pi$
$563$	−12789.8	−0.957417	−0.478709	+	0.877974i	$0.658895\pi$
$564$	0	0
$565$	998.832	0.0743737
$566$	−22129.9	−1.64345
$567$	0	0
$568$	7211.93	0.532757
$569$	10151.1	0.747904	0.373952	−	0.927448i	$-0.378002\pi$
$569$	10151.1	0.747904	0.373952	+	0.927448i	$0.378002\pi$
$570$	0	0
$571$	−3426.70	−0.251144	−0.125572	−	0.992085i	$-0.540077\pi$
$571$	−3426.70	−0.251144	−0.125572	+	0.992085i	$0.540077\pi$
$572$	1281.51	0.0936756
$573$	0	0
$574$	−16328.9	−1.18738
$575$	−20833.8	−1.51101
$576$	0	0
$577$	7548.20	0.544603	0.272301	−	0.962212i	$-0.412215\pi$
$577$	7548.20	0.544603	0.272301	+	0.962212i	$0.412215\pi$
$578$	−24239.6	−1.74435
$579$	0	0
$580$	1707.98	0.122276
$581$	13508.2	0.964568
$582$	0	0
$583$	731.655	0.0519761
$584$	15850.8	1.12313
$585$	0	0
$586$	16767.2	1.18199
$587$	−12736.8	−0.895577	−0.447789	−	0.894139i	$-0.647788\pi$
$587$	−12736.8	−0.895577	−0.447789	+	0.894139i	$0.647788\pi$
$588$	0	0
$589$	1619.44	0.113290
$590$	−2791.35	−0.194776
$591$	0	0
$592$	−11393.7	−0.791012
$593$	15108.1	1.04623	0.523117	−	0.852261i	$-0.324769\pi$
$593$	15108.1	1.04623	0.523117	+	0.852261i	$0.324769\pi$
$594$	0	0
$595$	2877.39	0.198255
$596$	14681.2	1.00900
$597$	0	0
$598$	14255.1	0.974808
$599$	4294.17	0.292913	0.146457	−	0.989217i	$-0.453213\pi$
$599$	4294.17	0.292913	0.146457	+	0.989217i	$0.453213\pi$
$600$	0	0
$601$	−10426.3	−0.707648	−0.353824	−	0.935312i	$-0.615119\pi$
$601$	−10426.3	−0.707648	−0.353824	+	0.935312i	$0.615119\pi$
$602$	−48386.5	−3.27589
$603$	0	0
$604$	49917.3	3.36276
$605$	−1670.10	−0.112230
$606$	0	0
$607$	−28473.2	−1.90394	−0.951971	−	0.306188i	$-0.900946\pi$
$607$	−28473.2	−1.90394	−0.951971	+	0.306188i	$0.900946\pi$
$608$	14786.3	0.986286
$609$	0	0
$610$	3334.82	0.221349
$611$	6269.42	0.415112
$612$	0	0
$613$	14942.9	0.984561	0.492281	−	0.870437i	$-0.336163\pi$
$613$	14942.9	0.984561	0.492281	+	0.870437i	$0.336163\pi$
$614$	−20177.7	−1.32623
$615$	0	0
$616$	−5295.31	−0.346354
$617$	11955.4	0.780072	0.390036	−	0.920800i	$-0.372463\pi$
$617$	11955.4	0.780072	0.390036	+	0.920800i	$0.372463\pi$
$618$	0	0
$619$	13024.9	0.845746	0.422873	−	0.906189i	$-0.361022\pi$
$619$	13024.9	0.845746	0.422873	+	0.906189i	$0.361022\pi$
$620$	−634.999	−0.0411325
$621$	0	0
$622$	−16106.2	−1.03826
$623$	13559.6	0.871995
$624$	0	0
$625$	15021.1	0.961352
$626$	−40795.6	−2.60467
$627$	0	0
$628$	−5551.76	−0.352769
$629$	8453.30	0.535859
$630$	0	0
$631$	−9384.17	−0.592041	−0.296021	−	0.955182i	$-0.595660\pi$
$631$	−9384.17	−0.592041	−0.296021	+	0.955182i	$0.595660\pi$
$632$	−51644.6	−3.25050
$633$	0	0
$634$	18103.5	1.13404
$635$	−1078.22	−0.0673826
$636$	0	0
$637$	−3101.78	−0.192931
$638$	1571.22	0.0975005
$639$	0	0
$640$	1132.96	0.0699751
$641$	−520.895	−0.0320969	−0.0160484	−	0.999871i	$-0.505109\pi$
$641$	−520.895	−0.0320969	−0.0160484	+	0.999871i	$0.505109\pi$
$642$	0	0
$643$	−24141.1	−1.48061	−0.740306	−	0.672270i	$-0.765320\pi$
$643$	−24141.1	−1.48061	−0.740306	+	0.672270i	$0.765320\pi$
$644$	−72333.8	−4.42601
$645$	0	0
$646$	−30434.5	−1.85360
$647$	8801.57	0.534815	0.267408	−	0.963583i	$-0.413833\pi$
$647$	8801.57	0.534815	0.267408	+	0.963583i	$0.413833\pi$
$648$	0	0
$649$	−1794.57	−0.108541
$650$	−10416.3	−0.628553
$651$	0	0
$652$	2759.42	0.165747
$653$	−12705.6	−0.761422	−0.380711	−	0.924694i	$-0.624321\pi$
$653$	−12705.6	−0.761422	−0.380711	+	0.924694i	$0.624321\pi$
$654$	0	0
$655$	−496.915	−0.0296429
$656$	−18147.6	−1.08010
$657$	0	0
$658$	−45520.3	−2.69691
$659$	−15876.5	−0.938483	−0.469241	−	0.883070i	$-0.655473\pi$
$659$	−15876.5	−0.938483	−0.469241	+	0.883070i	$0.655473\pi$
$660$	0	0
$661$	18295.7	1.07658	0.538291	−	0.842759i	$-0.319070\pi$
$661$	18295.7	1.07658	0.538291	+	0.842759i	$0.319070\pi$
$662$	−42952.8	−2.52176
$663$	0	0
$664$	31883.5	1.86343
$665$	1766.90	0.103034
$666$	0	0
$667$	12214.6	0.709075
$668$	60852.6	3.52463
$669$	0	0
$670$	4263.38	0.245834
$671$	2143.97	0.123349
$672$	0	0
$673$	20647.4	1.18261	0.591307	−	0.806446i	$-0.298612\pi$
$673$	20647.4	1.18261	0.591307	+	0.806446i	$0.298612\pi$
$674$	−24060.9	−1.37506
$675$	0	0
$676$	−35808.7	−2.03737
$677$	32567.1	1.84883	0.924413	−	0.381393i	$-0.124556\pi$
$677$	32567.1	1.84883	0.924413	+	0.381393i	$0.124556\pi$
$678$	0	0
$679$	11056.1	0.624879
$680$	6791.53	0.383005
$681$	0	0
$682$	−584.154	−0.0327983
$683$	5557.74	0.311363	0.155682	−	0.987807i	$-0.450243\pi$
$683$	5557.74	0.311363	0.155682	+	0.987807i	$0.450243\pi$
$684$	0	0
$685$	2882.28	0.160768
$686$	−18270.1	−1.01684
$687$	0	0
$688$	−53775.7	−2.97991
$689$	2843.76	0.157241
$690$	0	0
$691$	28353.5	1.56095	0.780477	−	0.625185i	$-0.214976\pi$
$691$	28353.5	1.56095	0.780477	+	0.625185i	$0.214976\pi$
$692$	−60593.2	−3.32862
$693$	0	0
$694$	22673.1	1.24015
$695$	−2405.20	−0.131273
$696$	0	0
$697$	13464.2	0.731695
$698$	−28918.3	−1.56816
$699$	0	0
$700$	52854.5	2.85387
$701$	4271.73	0.230159	0.115079	−	0.993356i	$-0.463288\pi$
$701$	4271.73	0.230159	0.115079	+	0.993356i	$0.463288\pi$
$702$	0	0
$703$	5190.87	0.278488
$704$	−877.758	−0.0469911
$705$	0	0
$706$	−502.327	−0.0267781
$707$	−39562.8	−2.10454
$708$	0	0
$709$	9667.95	0.512112	0.256056	−	0.966662i	$-0.417577\pi$
$709$	9667.95	0.512112	0.256056	+	0.966662i	$0.417577\pi$
$710$	868.031	0.0458826
$711$	0	0
$712$	32004.8	1.68459
$713$	−4541.19	−0.238526
$714$	0	0
$715$	87.7804	0.00459133
$716$	−83918.1	−4.38012
$717$	0	0
$718$	−41647.7	−2.16473
$719$	25456.1	1.32038	0.660188	−	0.751100i	$-0.270476\pi$
$719$	25456.1	1.32038	0.660188	+	0.751100i	$0.270476\pi$
$720$	0	0
$721$	−956.670	−0.0494151
$722$	16664.1	0.858968
$723$	0	0
$724$	−66843.2	−3.43122
$725$	−8925.27	−0.457209
$726$	0	0
$727$	6884.94	0.351235	0.175618	−	0.984458i	$-0.443808\pi$
$727$	6884.94	0.351235	0.175618	+	0.984458i	$0.443808\pi$
$728$	−20581.6	−1.04781
$729$	0	0
$730$	1907.80	0.0967274
$731$	39897.6	2.01870
$732$	0	0
$733$	96.5247	0.00486387	0.00243194	−	0.999997i	$-0.499226\pi$
$733$	96.5247	0.00486387	0.00243194	+	0.999997i	$0.499226\pi$
$734$	−25109.9	−1.26270
$735$	0	0
$736$	−41463.3	−2.07657
$737$	2740.95	0.136993
$738$	0	0
$739$	−18330.0	−0.912423	−0.456212	−	0.889871i	$-0.650794\pi$
$739$	−18330.0	−0.912423	−0.456212	+	0.889871i	$0.650794\pi$
$740$	−2035.39	−0.101112
$741$	0	0
$742$	−20647.7	−1.02156
$743$	−17703.9	−0.874152	−0.437076	−	0.899425i	$-0.643986\pi$
$743$	−17703.9	−0.874152	−0.437076	+	0.899425i	$0.643986\pi$
$744$	0	0
$745$	1005.63	0.0494544
$746$	36311.7	1.78213
$747$	0	0
$748$	7672.22	0.375032
$749$	22229.1	1.08442
$750$	0	0
$751$	20342.4	0.988421	0.494210	−	0.869342i	$-0.335457\pi$
$751$	20342.4	0.988421	0.494210	+	0.869342i	$0.335457\pi$
$752$	−50590.2	−2.45324
$753$	0	0
$754$	6106.95	0.294963
$755$	3419.23	0.164819
$756$	0	0
$757$	26846.7	1.28898	0.644490	−	0.764612i	$-0.277070\pi$
$757$	26846.7	1.28898	0.644490	+	0.764612i	$0.277070\pi$
$758$	−30767.6	−1.47431
$759$	0	0
$760$	4170.43	0.199049
$761$	1474.28	0.0702269	0.0351134	−	0.999383i	$-0.488821\pi$
$761$	1474.28	0.0702269	0.0351134	+	0.999383i	$0.488821\pi$
$762$	0	0
$763$	−11023.3	−0.523029
$764$	−29736.9	−1.40817
$765$	0	0
$766$	−18644.8	−0.879458
$767$	−6975.05	−0.328363
$768$	0	0
$769$	24129.6	1.13152	0.565758	−	0.824571i	$-0.308584\pi$
$769$	24129.6	1.13152	0.565758	+	0.824571i	$0.308584\pi$
$770$	−637.346	−0.0298290
$771$	0	0
$772$	−64229.7	−2.99440
$773$	−30272.0	−1.40855	−0.704273	−	0.709929i	$-0.748727\pi$
$773$	−30272.0	−1.40855	−0.704273	+	0.709929i	$0.748727\pi$
$774$	0	0
$775$	3318.26	0.153801
$776$	26095.7	1.20719
$777$	0	0
$778$	−16203.4	−0.746686
$779$	8267.86	0.380265
$780$	0	0
$781$	558.061	0.0255685
$782$	85343.7	3.90266
$783$	0	0
$784$	25029.4	1.14019
$785$	−380.283	−0.0172903
$786$	0	0
$787$	−30974.7	−1.40296	−0.701480	−	0.712690i	$-0.747477\pi$
$787$	−30974.7	−1.40296	−0.701480	+	0.712690i	$0.747477\pi$
$788$	15509.1	0.701128
$789$	0	0
$790$	−6215.97	−0.279942
$791$	−18121.9	−0.814589
$792$	0	0
$793$	8333.08	0.373161
$794$	−20419.2	−0.912657
$795$	0	0
$796$	−23221.5	−1.03400
$797$	9007.64	0.400335	0.200168	−	0.979762i	$-0.435851\pi$
$797$	9007.64	0.400335	0.200168	+	0.979762i	$0.435851\pi$
$798$	0	0
$799$	37534.2	1.66191
$800$	30297.3	1.33897
$801$	0	0
$802$	−26618.4	−1.17198
$803$	1226.54	0.0539023
$804$	0	0
$805$	−4954.71	−0.216932
$806$	−2270.46	−0.0992228
$807$	0	0
$808$	−93380.3	−4.06573
$809$	−12327.4	−0.535732	−0.267866	−	0.963456i	$-0.586318\pi$
$809$	−12327.4	−0.535732	−0.267866	+	0.963456i	$0.586318\pi$
$810$	0	0
$811$	−19319.9	−0.836514	−0.418257	−	0.908329i	$-0.637359\pi$
$811$	−19319.9	−0.836514	−0.418257	+	0.908329i	$0.637359\pi$
$812$	−30988.1	−1.33925
$813$	0	0
$814$	−1872.42	−0.0806243
$815$	189.014	0.00812379
$816$	0	0
$817$	24499.7	1.04913
$818$	78662.6	3.36231
$819$	0	0
$820$	−3241.91	−0.138064
$821$	−811.897	−0.0345133	−0.0172566	−	0.999851i	$-0.505493\pi$
$821$	−811.897	−0.0345133	−0.0172566	+	0.999851i	$0.505493\pi$
$822$	0	0
$823$	21030.5	0.890737	0.445369	−	0.895347i	$-0.353073\pi$
$823$	21030.5	0.890737	0.445369	+	0.895347i	$0.353073\pi$
$824$	−2258.04	−0.0954642
$825$	0	0
$826$	50643.6	2.13331
$827$	33934.2	1.42686	0.713428	−	0.700729i	$-0.247142\pi$
$827$	33934.2	1.42686	0.713428	+	0.700729i	$0.247142\pi$
$828$	0	0
$829$	21393.4	0.896289	0.448144	−	0.893961i	$-0.352085\pi$
$829$	21393.4	0.896289	0.448144	+	0.893961i	$0.352085\pi$
$830$	3837.51	0.160484
$831$	0	0
$832$	−3411.63	−0.142160
$833$	−18570.0	−0.772402
$834$	0	0
$835$	4168.27	0.172753
$836$	4711.24	0.194906
$837$	0	0
$838$	−45139.6	−1.86076
$839$	7747.11	0.318784	0.159392	−	0.987215i	$-0.449047\pi$
$839$	7747.11	0.318784	0.159392	+	0.987215i	$0.449047\pi$
$840$	0	0
$841$	−19156.2	−0.785444
$842$	15076.5	0.617066
$843$	0	0
$844$	17399.0	0.709597
$845$	−2452.82	−0.0998575
$846$	0	0
$847$	30300.7	1.22921
$848$	−22947.4	−0.929264
$849$	0	0
$850$	−62360.8	−2.51642
$851$	−14556.1	−0.586342
$852$	0	0
$853$	−16935.0	−0.679769	−0.339884	−	0.940467i	$-0.610388\pi$
$853$	−16935.0	−0.679769	−0.339884	+	0.940467i	$0.610388\pi$
$854$	−60503.9	−2.42436
$855$	0	0
$856$	52467.4	2.09498
$857$	3777.17	0.150555	0.0752776	−	0.997163i	$-0.476016\pi$
$857$	3777.17	0.150555	0.0752776	+	0.997163i	$0.476016\pi$
$858$	0	0
$859$	−21017.0	−0.834798	−0.417399	−	0.908723i	$-0.637058\pi$
$859$	−21017.0	−0.834798	−0.417399	+	0.908723i	$0.637058\pi$
$860$	−9606.59	−0.380909
$861$	0	0
$862$	27040.8	1.06846
$863$	−9505.46	−0.374936	−0.187468	−	0.982271i	$-0.560028\pi$
$863$	−9505.46	−0.374936	−0.187468	+	0.982271i	$0.560028\pi$
$864$	0	0
$865$	−4150.50	−0.163146
$866$	14955.2	0.586834
$867$	0	0
$868$	11520.8	0.450510
$869$	−3996.28	−0.156001
$870$	0	0
$871$	10653.4	0.414438
$872$	−26018.5	−1.01043
$873$	0	0
$874$	52406.5	2.02823
$875$	7288.29	0.281588
$876$	0	0
$877$	27293.6	1.05090	0.525450	−	0.850824i	$-0.323897\pi$
$877$	27293.6	1.05090	0.525450	+	0.850824i	$0.323897\pi$
$878$	−23542.1	−0.904906
$879$	0	0
$880$	−708.332	−0.0271340
$881$	28578.9	1.09290	0.546451	−	0.837491i	$-0.315978\pi$
$881$	28578.9	1.09290	0.546451	+	0.837491i	$0.315978\pi$
$882$	0	0
$883$	−5713.16	−0.217738	−0.108869	−	0.994056i	$-0.534723\pi$
$883$	−5713.16	−0.217738	−0.108869	+	0.994056i	$0.534723\pi$
$884$	29820.0	1.13457
$885$	0	0
$886$	10731.9	0.406934
$887$	−43896.4	−1.66166	−0.830832	−	0.556523i	$-0.812135\pi$
$887$	−43896.4	−1.66166	−0.830832	+	0.556523i	$0.812135\pi$
$888$	0	0
$889$	19562.3	0.738018
$890$	3852.11	0.145082
$891$	0	0
$892$	48079.9	1.80475
$893$	23048.4	0.863703
$894$	0	0
$895$	−5748.21	−0.214683
$896$	−20555.3	−0.766412
$897$	0	0
$898$	332.108	0.0123414
$899$	−1945.47	−0.0721746
$900$	0	0
$901$	17025.3	0.629515
$902$	−2982.33	−0.110089
$903$	0	0
$904$	−42773.2	−1.57369
$905$	−4578.61	−0.168175
$906$	0	0
$907$	23392.1	0.856362	0.428181	−	0.903693i	$-0.359155\pi$
$907$	23392.1	0.856362	0.428181	+	0.903693i	$0.359155\pi$
$908$	−95090.9	−3.47544
$909$	0	0
$910$	−2477.21	−0.0902402
$911$	−1315.19	−0.0478310	−0.0239155	−	0.999714i	$-0.507613\pi$
$911$	−1315.19	−0.0478310	−0.0239155	+	0.999714i	$0.507613\pi$
$912$	0	0
$913$	2467.15	0.0894314
$914$	44358.0	1.60529
$915$	0	0
$916$	66658.5	2.40443
$917$	9015.57	0.324668
$918$	0	0
$919$	−47277.7	−1.69701	−0.848503	−	0.529191i	$-0.822496\pi$
$919$	−47277.7	−1.69701	−0.848503	+	0.529191i	$0.822496\pi$
$920$	−11694.6	−0.419088
$921$	0	0
$922$	−79453.7	−2.83804
$923$	2169.05	0.0773510
$924$	0	0
$925$	10636.2	0.378071
$926$	−54800.9	−1.94478
$927$	0	0
$928$	−17763.0	−0.628341
$929$	−36239.8	−1.27986	−0.639930	−	0.768433i	$-0.721037\pi$
$929$	−36239.8	−1.27986	−0.639930	+	0.768433i	$0.721037\pi$
$930$	0	0
$931$	−11403.2	−0.401421
$932$	−16407.4	−0.576654
$933$	0	0
$934$	13983.3	0.489879
$935$	525.531	0.0183815
$936$	0	0
$937$	12678.9	0.442051	0.221026	−	0.975268i	$-0.429060\pi$
$937$	12678.9	0.442051	0.221026	+	0.975268i	$0.429060\pi$
$938$	−77350.8	−2.69253
$939$	0	0
$940$	−9037.53	−0.313587
$941$	50353.2	1.74438	0.872192	−	0.489163i	$-0.162698\pi$
$941$	50353.2	1.74438	0.872192	+	0.489163i	$0.162698\pi$
$942$	0	0
$943$	−23184.5	−0.800628
$944$	56284.2	1.94057
$945$	0	0
$946$	−8837.38	−0.303729
$947$	−33033.7	−1.13353	−0.566763	−	0.823881i	$-0.691805\pi$
$947$	−33033.7	−1.13353	−0.566763	+	0.823881i	$0.691805\pi$
$948$	0	0
$949$	4767.24	0.163068
$950$	−38293.6	−1.30780
$951$	0	0
$952$	−123219.	−4.19492
$953$	−12625.3	−0.429142	−0.214571	−	0.976708i	$-0.568835\pi$
$953$	−12625.3	−0.429142	−0.214571	+	0.976708i	$0.568835\pi$
$954$	0	0
$955$	−2036.92	−0.0690189
$956$	−4437.29	−0.150117
$957$	0	0
$958$	−49880.8	−1.68223
$959$	−52293.4	−1.76083
$960$	0	0
$961$	−29067.7	−0.975721
$962$	−7277.62	−0.243909
$963$	0	0
$964$	−26866.7	−0.897633
$965$	−4399.59	−0.146765
$966$	0	0
$967$	−20457.9	−0.680333	−0.340167	−	0.940365i	$-0.610484\pi$
$967$	−20457.9	−0.680333	−0.340167	+	0.940365i	$0.610484\pi$
$968$	71519.0	2.37470
$969$	0	0
$970$	3140.89	0.103967
$971$	41805.4	1.38167	0.690834	−	0.723014i	$-0.257244\pi$
$971$	41805.4	1.38167	0.690834	+	0.723014i	$0.257244\pi$
$972$	0	0
$973$	43637.8	1.43778
$974$	7518.31	0.247333
$975$	0	0
$976$	−67242.7	−2.20531
$977$	47505.6	1.55562	0.777808	−	0.628502i	$-0.216331\pi$
$977$	47505.6	1.55562	0.777808	+	0.628502i	$0.216331\pi$
$978$	0	0
$979$	2476.54	0.0808484
$980$	4471.30	0.145745
$981$	0	0
$982$	54108.0	1.75830
$983$	58004.1	1.88204	0.941020	−	0.338352i	$-0.109869\pi$
$983$	58004.1	1.88204	0.941020	+	0.338352i	$0.109869\pi$
$984$	0	0
$985$	1062.34	0.0343644
$986$	36561.6	1.18089
$987$	0	0
$988$	18311.4	0.589639
$989$	−68701.5	−2.20888
$990$	0	0
$991$	46228.7	1.48184	0.740920	−	0.671593i	$-0.234390\pi$
$991$	46228.7	1.48184	0.740920	+	0.671593i	$0.234390\pi$
$992$	6604.00	0.211368
$993$	0	0
$994$	−15748.8	−0.502535
$995$	−1590.62	−0.0506796
$996$	0	0
$997$	−35226.8	−1.11900	−0.559501	−	0.828830i	$-0.689007\pi$
$997$	−35226.8	−1.11900	−0.559501	+	0.828830i	$0.689007\pi$
$998$	−97494.4	−3.09232
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.h.1.3	yes	59
3.2	odd	2		2151.4.a.g.1.57	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.57	✓	59	3.2	odd	2
2151.4.a.h.1.3	yes	59	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)

\(f(q)\)	\(=\)	\(q-5.15423 q^{2} +18.5661 q^{4} +1.27174 q^{5} -23.0732 q^{7} -54.4599 q^{8} +O(q^{10})\)	\(q-5.15423 q^{2} +18.5661 q^{4} +1.27174 q^{5} -23.0732 q^{7} -54.4599 q^{8} -6.55481 q^{10} -4.21412 q^{11} -16.3792 q^{13} +118.925 q^{14} +132.170 q^{16} -98.0605 q^{17} -60.2154 q^{19} +23.6111 q^{20} +21.7205 q^{22} +168.855 q^{23} -123.383 q^{25} +84.4224 q^{26} -428.378 q^{28} +72.3381 q^{29} -26.8941 q^{31} -245.556 q^{32} +505.426 q^{34} -29.3430 q^{35} -86.2049 q^{37} +310.364 q^{38} -69.2585 q^{40} -137.305 q^{41} -406.867 q^{43} -78.2396 q^{44} -870.316 q^{46} -382.766 q^{47} +189.373 q^{49} +635.942 q^{50} -304.098 q^{52} -173.620 q^{53} -5.35925 q^{55} +1256.56 q^{56} -372.847 q^{58} +425.847 q^{59} -508.759 q^{61} +138.618 q^{62} +208.290 q^{64} -20.8301 q^{65} -650.419 q^{67} -1820.60 q^{68} +151.241 q^{70} -132.426 q^{71} -291.054 q^{73} +444.320 q^{74} -1117.96 q^{76} +97.2333 q^{77} +948.306 q^{79} +168.085 q^{80} +707.699 q^{82} -585.449 q^{83} -124.707 q^{85} +2097.09 q^{86} +229.501 q^{88} -587.676 q^{89} +377.922 q^{91} +3134.97 q^{92} +1972.86 q^{94} -76.5781 q^{95} -479.173 q^{97} -976.069 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) \) 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8} - 36 q^{10} + 132 q^{11} + 104 q^{13} + 280 q^{14} + 822 q^{16} + 408 q^{17} + 20 q^{19} + 800 q^{20} - 2 q^{22} + 276 q^{23} + 1477 q^{25} + 780 q^{26} + 224 q^{28} + 696 q^{29} - 380 q^{31} + 896 q^{32} - 72 q^{34} + 700 q^{35} + 224 q^{37} + 988 q^{38} - 258 q^{40} + 2706 q^{41} - 156 q^{43} + 1584 q^{44} + 428 q^{46} + 1316 q^{47} + 2135 q^{49} + 1400 q^{50} + 1092 q^{52} + 1484 q^{53} - 992 q^{55} + 3360 q^{56} - 120 q^{58} + 3186 q^{59} - 254 q^{61} + 1240 q^{62} + 3054 q^{64} + 5120 q^{65} + 288 q^{67} + 9420 q^{68} + 1108 q^{70} + 4468 q^{71} - 1770 q^{73} + 6214 q^{74} + 720 q^{76} + 6352 q^{77} - 746 q^{79} + 7040 q^{80} + 276 q^{82} + 5484 q^{83} + 588 q^{85} + 10152 q^{86} + 1186 q^{88} + 11570 q^{89} + 1768 q^{91} + 15366 q^{92} - 2142 q^{94} + 5736 q^{95} + 2390 q^{97} + 6912 q^{98}+O(q^{100}) \) 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8 - 36 * q^10 + 132 * q^11 + 104 * q^13 + 280 * q^14 + 822 * q^16 + 408 * q^17 + 20 * q^19 + 800 * q^20 - 2 * q^22 + 276 * q^23 + 1477 * q^25 + 780 * q^26 + 224 * q^28 + 696 * q^29 - 380 * q^31 + 896 * q^32 - 72 * q^34 + 700 * q^35 + 224 * q^37 + 988 * q^38 - 258 * q^40 + 2706 * q^41 - 156 * q^43 + 1584 * q^44 + 428 * q^46 + 1316 * q^47 + 2135 * q^49 + 1400 * q^50 + 1092 * q^52 + 1484 * q^53 - 992 * q^55 + 3360 * q^56 - 120 * q^58 + 3186 * q^59 - 254 * q^61 + 1240 * q^62 + 3054 * q^64 + 5120 * q^65 + 288 * q^67 + 9420 * q^68 + 1108 * q^70 + 4468 * q^71 - 1770 * q^73 + 6214 * q^74 + 720 * q^76 + 6352 * q^77 - 746 * q^79 + 7040 * q^80 + 276 * q^82 + 5484 * q^83 + 588 * q^85 + 10152 * q^86 + 1186 * q^88 + 11570 * q^89 + 1768 * q^91 + 15366 * q^92 - 2142 * q^94 + 5736 * q^95 + 2390 * q^97 + 6912 * q^98

Introduction

L-functions

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