LMFDB - Embedded newform 2151.4.a.h.1.18

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			1.18
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)

$f(q)$	$=$	$q-2.78319 q^{2} -0.253830 q^{4} +19.2360 q^{5} +27.5534 q^{7} +22.9720 q^{8} +O(q^{10})$	\(q-2.78319 q^{2} -0.253830 q^{4} +19.2360 q^{5} +27.5534 q^{7} +22.9720 q^{8} -53.5375 q^{10} +32.2259 q^{11} +10.3407 q^{13} -76.6866 q^{14} -61.9049 q^{16} +38.8792 q^{17} +137.845 q^{19} -4.88268 q^{20} -89.6910 q^{22} -158.387 q^{23} +245.024 q^{25} -28.7802 q^{26} -6.99390 q^{28} -30.8395 q^{29} +35.0179 q^{31} -11.4827 q^{32} -108.208 q^{34} +530.018 q^{35} +105.071 q^{37} -383.651 q^{38} +441.890 q^{40} +295.652 q^{41} -311.117 q^{43} -8.17992 q^{44} +440.823 q^{46} -122.518 q^{47} +416.192 q^{49} -681.949 q^{50} -2.62478 q^{52} +700.716 q^{53} +619.898 q^{55} +632.958 q^{56} +85.8324 q^{58} -548.102 q^{59} +206.571 q^{61} -97.4616 q^{62} +527.198 q^{64} +198.914 q^{65} +352.913 q^{67} -9.86873 q^{68} -1475.14 q^{70} +759.373 q^{71} -587.570 q^{73} -292.434 q^{74} -34.9894 q^{76} +887.935 q^{77} -149.398 q^{79} -1190.80 q^{80} -822.857 q^{82} +1024.10 q^{83} +747.881 q^{85} +865.900 q^{86} +740.295 q^{88} +1045.70 q^{89} +284.922 q^{91} +40.2035 q^{92} +340.991 q^{94} +2651.60 q^{95} -1808.66 q^{97} -1158.34 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.78319	−0.984008	−0.492004	−	0.870593i	$-0.663735\pi$
$2$	−2.78319	−0.984008	−0.492004	+	0.870593i	$0.663735\pi$
$3$	0	0
$3$	0	0
$4$	−0.253830	−0.0317288
$5$	19.2360	1.72052	0.860260	−	0.509855i	$-0.170301\pi$
$5$	19.2360	1.72052	0.860260	+	0.509855i	$0.170301\pi$
$6$	0	0
$7$	27.5534	1.48775	0.743873	−	0.668321i	$-0.232987\pi$
$7$	27.5534	1.48775	0.743873	+	0.668321i	$0.232987\pi$
$8$	22.9720	1.01523
$9$	0	0
$10$	−53.5375	−1.69301
$11$	32.2259	0.883317	0.441658	−	0.897183i	$-0.354390\pi$
$11$	32.2259	0.883317	0.441658	+	0.897183i	$0.354390\pi$
$12$	0	0
$13$	10.3407	0.220615	0.110307	−	0.993898i	$-0.464816\pi$
$13$	10.3407	0.220615	0.110307	+	0.993898i	$0.464816\pi$
$14$	−76.6866	−1.46395
$15$	0	0
$16$	−61.9049	−0.967264
$17$	38.8792	0.554682	0.277341	−	0.960771i	$-0.410547\pi$
$17$	38.8792	0.554682	0.277341	+	0.960771i	$0.410547\pi$
$18$	0	0
$19$	137.845	1.66442	0.832208	−	0.554463i	$-0.187076\pi$
$19$	137.845	1.66442	0.832208	+	0.554463i	$0.187076\pi$
$20$	−4.88268	−0.0545901
$21$	0	0
$22$	−89.6910	−0.869191
$23$	−158.387	−1.43592	−0.717958	−	0.696087i	$-0.754923\pi$
$23$	−158.387	−1.43592	−0.717958	+	0.696087i	$0.754923\pi$
$24$	0	0
$25$	245.024	1.96019
$26$	−28.7802	−0.217087
$27$	0	0
$28$	−6.99390	−0.0472044
$29$	−30.8395	−0.197474	−0.0987372	−	0.995114i	$-0.531480\pi$
$29$	−30.8395	−0.197474	−0.0987372	+	0.995114i	$0.531480\pi$
$30$	0	0
$31$	35.0179	0.202884	0.101442	−	0.994841i	$-0.467654\pi$
$31$	35.0179	0.202884	0.101442	+	0.994841i	$0.467654\pi$
$32$	−11.4827	−0.0634334
$33$	0	0
$34$	−108.208	−0.545812
$35$	530.018	2.55970
$36$	0	0
$37$	105.071	0.466855	0.233427	−	0.972374i	$-0.425006\pi$
$37$	105.071	0.466855	0.233427	+	0.972374i	$0.425006\pi$
$38$	−383.651	−1.63780
$39$	0	0
$40$	441.890	1.74672
$41$	295.652	1.12617	0.563086	−	0.826398i	$-0.309614\pi$
$41$	295.652	1.12617	0.563086	+	0.826398i	$0.309614\pi$
$42$	0	0
$43$	−311.117	−1.10337	−0.551686	−	0.834052i	$-0.686015\pi$
$43$	−311.117	−1.10337	−0.551686	+	0.834052i	$0.686015\pi$
$44$	−8.17992	−0.0280266
$45$	0	0
$46$	440.823	1.41295
$47$	−122.518	−0.380236	−0.190118	−	0.981761i	$-0.560887\pi$
$47$	−122.518	−0.380236	−0.190118	+	0.981761i	$0.560887\pi$
$48$	0	0
$49$	416.192	1.21339
$50$	−681.949	−1.92884
$51$	0	0
$52$	−2.62478	−0.00699984
$53$	700.716	1.81605	0.908025	−	0.418915i	$-0.137590\pi$
$53$	700.716	1.81605	0.908025	+	0.418915i	$0.137590\pi$
$54$	0	0
$55$	619.898	1.51977
$56$	632.958	1.51040
$57$	0	0
$58$	85.8324	0.194316
$59$	−548.102	−1.20944	−0.604718	−	0.796439i	$-0.706714\pi$
$59$	−548.102	−1.20944	−0.604718	+	0.796439i	$0.706714\pi$
$60$	0	0
$61$	206.571	0.433586	0.216793	−	0.976218i	$-0.430440\pi$
$61$	206.571	0.433586	0.216793	+	0.976218i	$0.430440\pi$
$62$	−97.4616	−0.199639
$63$	0	0
$64$	527.198	1.02968
$65$	198.914	0.379573
$66$	0	0
$67$	352.913	0.643511	0.321755	−	0.946823i	$-0.395727\pi$
$67$	352.913	0.643511	0.321755	+	0.946823i	$0.395727\pi$
$68$	−9.86873	−0.0175994
$69$	0	0
$70$	−1475.14	−2.51876
$71$	759.373	1.26931	0.634655	−	0.772796i	$-0.281142\pi$
$71$	759.373	1.26931	0.634655	+	0.772796i	$0.281142\pi$
$72$	0	0
$73$	−587.570	−0.942054	−0.471027	−	0.882119i	$-0.656116\pi$
$73$	−587.570	−0.942054	−0.471027	+	0.882119i	$0.656116\pi$
$74$	−292.434	−0.459389
$75$	0	0
$76$	−34.9894	−0.0528099
$77$	887.935	1.31415
$78$	0	0
$79$	−149.398	−0.212767	−0.106384	−	0.994325i	$-0.533927\pi$
$79$	−149.398	−0.212767	−0.106384	+	0.994325i	$0.533927\pi$
$80$	−1190.80	−1.66420
$81$	0	0
$82$	−822.857	−1.10816
$83$	1024.10	1.35434	0.677168	−	0.735829i	$-0.263207\pi$
$83$	1024.10	1.35434	0.677168	+	0.735829i	$0.263207\pi$
$84$	0	0
$85$	747.881	0.954343
$86$	865.900	1.08573
$87$	0	0
$88$	740.295	0.896769
$89$	1045.70	1.24544	0.622718	−	0.782447i	$-0.286029\pi$
$89$	1045.70	1.24544	0.622718	+	0.782447i	$0.286029\pi$
$90$	0	0
$91$	284.922	0.328219
$92$	40.2035	0.0455599
$93$	0	0
$94$	340.991	0.374155
$95$	2651.60	2.86366
$96$	0	0
$97$	−1808.66	−1.89321	−0.946607	−	0.322390i	$-0.895514\pi$
$97$	−1808.66	−1.89321	−0.946607	+	0.322390i	$0.895514\pi$
$98$	−1158.34	−1.19398
$99$	0	0
$100$	−62.1945	−0.0621945
$101$	940.319	0.926388	0.463194	−	0.886257i	$-0.346703\pi$
$101$	940.319	0.926388	0.463194	+	0.886257i	$0.346703\pi$
$102$	0	0
$103$	−105.854	−0.101263	−0.0506314	−	0.998717i	$-0.516123\pi$
$103$	−105.854	−0.101263	−0.0506314	+	0.998717i	$0.516123\pi$
$104$	237.547	0.223975
$105$	0	0
$106$	−1950.23	−1.78701
$107$	−1946.90	−1.75901	−0.879506	−	0.475887i	$-0.842127\pi$
$107$	−1946.90	−1.75901	−0.879506	+	0.475887i	$0.842127\pi$
$108$	0	0
$109$	1837.70	1.61486	0.807431	−	0.589961i	$-0.200857\pi$
$109$	1837.70	1.61486	0.807431	+	0.589961i	$0.200857\pi$
$110$	−1725.30	−1.49546
$111$	0	0
$112$	−1705.69	−1.43904
$113$	−1446.76	−1.20442	−0.602212	−	0.798336i	$-0.705714\pi$
$113$	−1446.76	−1.20442	−0.602212	+	0.798336i	$0.705714\pi$
$114$	0	0
$115$	−3046.74	−2.47052
$116$	7.82801	0.00626563
$117$	0	0
$118$	1525.47	1.19010
$119$	1071.26	0.825227
$120$	0	0
$121$	−292.489	−0.219751
$122$	−574.928	−0.426652
$123$	0	0
$124$	−8.88860	−0.00643726
$125$	2308.78	1.65203
$126$	0	0
$127$	−1939.30	−1.35500	−0.677499	−	0.735524i	$-0.736936\pi$
$127$	−1939.30	−1.35500	−0.677499	+	0.735524i	$0.736936\pi$
$128$	−1375.43	−0.949783
$129$	0	0
$130$	−553.616	−0.373502
$131$	−501.024	−0.334158	−0.167079	−	0.985944i	$-0.553433\pi$
$131$	−501.024	−0.334158	−0.167079	+	0.985944i	$0.553433\pi$
$132$	0	0
$133$	3798.12	2.47623
$134$	−982.226	−0.633220
$135$	0	0
$136$	893.134	0.563130
$137$	−2450.94	−1.52845	−0.764227	−	0.644947i	$-0.776879\pi$
$137$	−2450.94	−1.52845	−0.764227	+	0.644947i	$0.776879\pi$
$138$	0	0
$139$	257.133	0.156905	0.0784523	−	0.996918i	$-0.475002\pi$
$139$	257.133	0.156905	0.0784523	+	0.996918i	$0.475002\pi$
$140$	−134.535	−0.0812161
$141$	0	0
$142$	−2113.48	−1.24901
$143$	333.239	0.194873
$144$	0	0
$145$	−593.230	−0.339759
$146$	1635.32	0.926988
$147$	0	0
$148$	−26.6703	−0.0148127
$149$	2187.75	1.20287	0.601434	−	0.798923i	$-0.294596\pi$
$149$	2187.75	1.20287	0.601434	+	0.798923i	$0.294596\pi$
$150$	0	0
$151$	−1250.56	−0.673970	−0.336985	−	0.941510i	$-0.609407\pi$
$151$	−1250.56	−0.673970	−0.336985	+	0.941510i	$0.609407\pi$
$152$	3166.59	1.68976
$153$	0	0
$154$	−2471.30	−1.29313
$155$	673.604	0.349066
$156$	0	0
$157$	−1312.84	−0.667363	−0.333682	−	0.942686i	$-0.608291\pi$
$157$	−1312.84	−0.667363	−0.333682	+	0.942686i	$0.608291\pi$
$158$	415.804	0.209365
$159$	0	0
$160$	−220.881	−0.109138
$161$	−4364.12	−2.13628
$162$	0	0
$163$	−1766.12	−0.848670	−0.424335	−	0.905505i	$-0.639492\pi$
$163$	−1766.12	−0.848670	−0.424335	+	0.905505i	$0.639492\pi$
$164$	−75.0454	−0.0357321
$165$	0	0
$166$	−2850.28	−1.33268
$167$	780.417	0.361620	0.180810	−	0.983518i	$-0.442128\pi$
$167$	780.417	0.361620	0.180810	+	0.983518i	$0.442128\pi$
$168$	0	0
$169$	−2090.07	−0.951329
$170$	−2081.50	−0.939081
$171$	0	0
$172$	78.9711	0.0350086
$173$	−3049.96	−1.34037	−0.670185	−	0.742194i	$-0.733785\pi$
$173$	−3049.96	−1.34037	−0.670185	+	0.742194i	$0.733785\pi$
$174$	0	0
$175$	6751.26	2.91627
$176$	−1994.94	−0.854401
$177$	0	0
$178$	−2910.38	−1.22552
$179$	−1950.16	−0.814312	−0.407156	−	0.913359i	$-0.633480\pi$
$179$	−1950.16	−0.814312	−0.407156	+	0.913359i	$0.633480\pi$
$180$	0	0
$181$	−2597.95	−1.06687	−0.533436	−	0.845840i	$-0.679099\pi$
$181$	−2597.95	−1.06687	−0.533436	+	0.845840i	$0.679099\pi$
$182$	−792.993	−0.322970
$183$	0	0
$184$	−3638.48	−1.45778
$185$	2021.15	0.803234
$186$	0	0
$187$	1252.92	0.489960
$188$	31.0988	0.0120644
$189$	0	0
$190$	−7379.91	−2.81787
$191$	1447.23	0.548260	0.274130	−	0.961693i	$-0.411610\pi$
$191$	1447.23	0.548260	0.274130	+	0.961693i	$0.411610\pi$
$192$	0	0
$193$	−1220.05	−0.455033	−0.227516	−	0.973774i	$-0.573061\pi$
$193$	−1220.05	−0.455033	−0.227516	+	0.973774i	$0.573061\pi$
$194$	5033.86	1.86294
$195$	0	0
$196$	−105.642	−0.0384994
$197$	2862.29	1.03518	0.517588	−	0.855630i	$-0.326830\pi$
$197$	2862.29	1.03518	0.517588	+	0.855630i	$0.326830\pi$
$198$	0	0
$199$	−267.746	−0.0953771	−0.0476886	−	0.998862i	$-0.515185\pi$
$199$	−267.746	−0.0953771	−0.0476886	+	0.998862i	$0.515185\pi$
$200$	5628.70	1.99004
$201$	0	0
$202$	−2617.09	−0.911573
$203$	−849.736	−0.293792
$204$	0	0
$205$	5687.16	1.93760
$206$	294.611	0.0996434
$207$	0	0
$208$	−640.140	−0.213393
$209$	4442.20	1.47021
$210$	0	0
$211$	−2491.09	−0.812766	−0.406383	−	0.913703i	$-0.633210\pi$
$211$	−2491.09	−0.812766	−0.406383	+	0.913703i	$0.633210\pi$
$212$	−177.863	−0.0576211
$213$	0	0
$214$	5418.61	1.73088
$215$	−5984.66	−1.89837
$216$	0	0
$217$	964.863	0.301840
$218$	−5114.69	−1.58904
$219$	0	0
$220$	−157.349	−0.0482203
$221$	402.038	0.122371
$222$	0	0
$223$	5470.72	1.64281	0.821405	−	0.570346i	$-0.193191\pi$
$223$	5470.72	1.64281	0.821405	+	0.570346i	$0.193191\pi$
$224$	−316.387	−0.0943728
$225$	0	0
$226$	4026.62	1.18516
$227$	4177.85	1.22156	0.610779	−	0.791801i	$-0.290856\pi$
$227$	4177.85	1.22156	0.610779	+	0.791801i	$0.290856\pi$
$228$	0	0
$229$	3049.05	0.879857	0.439928	−	0.898033i	$-0.355004\pi$
$229$	3049.05	0.879857	0.439928	+	0.898033i	$0.355004\pi$
$230$	8479.67	2.43101
$231$	0	0
$232$	−708.446	−0.200482
$233$	265.914	0.0747666	0.0373833	−	0.999301i	$-0.488098\pi$
$233$	265.914	0.0747666	0.0373833	+	0.999301i	$0.488098\pi$
$234$	0	0
$235$	−2356.76	−0.654203
$236$	139.125	0.0383740
$237$	0	0
$238$	−2981.52	−0.812029
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−6888.63	−1.84123	−0.920614	−	0.390475i	$-0.872311\pi$
$241$	−6888.63	−1.84123	−0.920614	+	0.390475i	$0.872311\pi$
$242$	814.054	0.216237
$243$	0	0
$244$	−52.4340	−0.0137571
$245$	8005.88	2.08766
$246$	0	0
$247$	1425.42	0.367195
$248$	804.431	0.205973
$249$	0	0
$250$	−6425.79	−1.62561
$251$	−1646.85	−0.414136	−0.207068	−	0.978327i	$-0.566392\pi$
$251$	−1646.85	−0.414136	−0.207068	+	0.978327i	$0.566392\pi$
$252$	0	0
$253$	−5104.18	−1.26837
$254$	5397.44	1.33333
$255$	0	0
$256$	−389.487	−0.0950895
$257$	−5771.91	−1.40094	−0.700471	−	0.713681i	$-0.747027\pi$
$257$	−5771.91	−1.40094	−0.700471	+	0.713681i	$0.747027\pi$
$258$	0	0
$259$	2895.08	0.694562
$260$	−50.4904	−0.0120434
$261$	0	0
$262$	1394.45	0.328814
$263$	1656.33	0.388341	0.194171	−	0.980968i	$-0.437798\pi$
$263$	1656.33	0.388341	0.194171	+	0.980968i	$0.437798\pi$
$264$	0	0
$265$	13479.0	3.12455
$266$	−10570.9	−2.43663
$267$	0	0
$268$	−89.5801	−0.0204178
$269$	2471.45	0.560174	0.280087	−	0.959975i	$-0.409637\pi$
$269$	2471.45	0.560174	0.280087	+	0.959975i	$0.409637\pi$
$270$	0	0
$271$	44.2396	0.00991648	0.00495824	−	0.999988i	$-0.498422\pi$
$271$	44.2396	0.00991648	0.00495824	+	0.999988i	$0.498422\pi$
$272$	−2406.82	−0.536525
$273$	0	0
$274$	6821.45	1.50401
$275$	7896.13	1.73147
$276$	0	0
$277$	−2924.86	−0.634433	−0.317217	−	0.948353i	$-0.602748\pi$
$277$	−2924.86	−0.634433	−0.317217	+	0.948353i	$0.602748\pi$
$278$	−715.651	−0.154395
$279$	0	0
$280$	12175.6	2.59868
$281$	7081.71	1.50341	0.751707	−	0.659497i	$-0.229231\pi$
$281$	7081.71	1.50341	0.751707	+	0.659497i	$0.229231\pi$
$282$	0	0
$283$	473.768	0.0995143	0.0497572	−	0.998761i	$-0.484155\pi$
$283$	473.768	0.0995143	0.0497572	+	0.998761i	$0.484155\pi$
$284$	−192.752	−0.0402737
$285$	0	0
$286$	−927.468	−0.191756
$287$	8146.23	1.67546
$288$	0	0
$289$	−3401.40	−0.692327
$290$	1651.07	0.334325
$291$	0	0
$292$	149.143	0.0298902
$293$	9805.59	1.95512	0.977558	−	0.210668i	$-0.0675637\pi$
$293$	9805.59	1.95512	0.977558	+	0.210668i	$0.0675637\pi$
$294$	0	0
$295$	−10543.3	−2.08086
$296$	2413.70	0.473965
$297$	0	0
$298$	−6088.92	−1.18363
$299$	−1637.84	−0.316784
$300$	0	0
$301$	−8572.36	−1.64154
$302$	3480.56	0.663191
$303$	0	0
$304$	−8533.31	−1.60993
$305$	3973.60	0.745993
$306$	0	0
$307$	−1384.15	−0.257321	−0.128660	−	0.991689i	$-0.541068\pi$
$307$	−1384.15	−0.257321	−0.128660	+	0.991689i	$0.541068\pi$
$308$	−225.385	−0.0416964
$309$	0	0
$310$	−1874.77	−0.343483
$311$	−1774.27	−0.323504	−0.161752	−	0.986831i	$-0.551715\pi$
$311$	−1774.27	−0.323504	−0.161752	+	0.986831i	$0.551715\pi$
$312$	0	0
$313$	3700.09	0.668184	0.334092	−	0.942540i	$-0.391570\pi$
$313$	3700.09	0.668184	0.334092	+	0.942540i	$0.391570\pi$
$314$	3653.89	0.656690
$315$	0	0
$316$	37.9218	0.00675085
$317$	−3471.60	−0.615093	−0.307547	−	0.951533i	$-0.599508\pi$
$317$	−3471.60	−0.615093	−0.307547	+	0.951533i	$0.599508\pi$
$318$	0	0
$319$	−993.833	−0.174433
$320$	10141.2	1.77159
$321$	0	0
$322$	12146.2	2.10211
$323$	5359.33	0.923222
$324$	0	0
$325$	2533.72	0.432448
$326$	4915.45	0.835097
$327$	0	0
$328$	6791.72	1.14332
$329$	−3375.79	−0.565694
$330$	0	0
$331$	−11769.1	−1.95434	−0.977172	−	0.212451i	$-0.931855\pi$
$331$	−11769.1	−1.95434	−0.977172	+	0.212451i	$0.931855\pi$
$332$	−259.948	−0.0429714
$333$	0	0
$334$	−2172.05	−0.355837
$335$	6788.64	1.10717
$336$	0	0
$337$	−10992.7	−1.77689	−0.888444	−	0.458985i	$-0.848213\pi$
$337$	−10992.7	−1.77689	−0.888444	+	0.458985i	$0.848213\pi$
$338$	5817.07	0.936115
$339$	0	0
$340$	−189.835	−0.0302801
$341$	1128.48	0.179211
$342$	0	0
$343$	2016.70	0.317468
$344$	−7146.99	−1.12017
$345$	0	0
$346$	8488.63	1.31893
$347$	6679.54	1.03336	0.516681	−	0.856178i	$-0.327167\pi$
$347$	6679.54	1.03336	0.516681	+	0.856178i	$0.327167\pi$
$348$	0	0
$349$	7868.92	1.20692	0.603458	−	0.797395i	$-0.293789\pi$
$349$	7868.92	1.20692	0.603458	+	0.797395i	$0.293789\pi$
$350$	−18790.1	−2.86963
$351$	0	0
$352$	−370.040	−0.0560318
$353$	−8705.88	−1.31266	−0.656328	−	0.754476i	$-0.727891\pi$
$353$	−8705.88	−1.31266	−0.656328	+	0.754476i	$0.727891\pi$
$354$	0	0
$355$	14607.3	2.18387
$356$	−265.430	−0.0395162
$357$	0	0
$358$	5427.68	0.801290
$359$	−1128.36	−0.165885	−0.0829425	−	0.996554i	$-0.526432\pi$
$359$	−1128.36	−0.165885	−0.0829425	+	0.996554i	$0.526432\pi$
$360$	0	0
$361$	12142.4	1.77028
$362$	7230.59	1.04981
$363$	0	0
$364$	−72.3218	−0.0104140
$365$	−11302.5	−1.62082
$366$	0	0
$367$	−2218.05	−0.315480	−0.157740	−	0.987481i	$-0.550421\pi$
$367$	−2218.05	−0.315480	−0.157740	+	0.987481i	$0.550421\pi$
$368$	9804.96	1.38891
$369$	0	0
$370$	−5625.26	−0.790388
$371$	19307.1	2.70182
$372$	0	0
$373$	−8888.30	−1.23383	−0.616915	−	0.787030i	$-0.711618\pi$
$373$	−8888.30	−1.23383	−0.616915	+	0.787030i	$0.711618\pi$
$374$	−3487.12	−0.482125
$375$	0	0
$376$	−2814.48	−0.386026
$377$	−318.902	−0.0435658
$378$	0	0
$379$	10662.5	1.44511	0.722556	−	0.691312i	$-0.242967\pi$
$379$	10662.5	1.44511	0.722556	+	0.691312i	$0.242967\pi$
$380$	−673.056	−0.0908606
$381$	0	0
$382$	−4027.91	−0.539492
$383$	3150.75	0.420354	0.210177	−	0.977663i	$-0.432596\pi$
$383$	3150.75	0.420354	0.210177	+	0.977663i	$0.432596\pi$
$384$	0	0
$385$	17080.3	2.26102
$386$	3395.65	0.447756
$387$	0	0
$388$	459.093	0.0600694
$389$	−11728.0	−1.52862	−0.764311	−	0.644848i	$-0.776921\pi$
$389$	−11728.0	−1.52862	−0.764311	+	0.644848i	$0.776921\pi$
$390$	0	0
$391$	−6157.98	−0.796477
$392$	9560.77	1.23187
$393$	0	0
$394$	−7966.31	−1.01862
$395$	−2873.83	−0.366070
$396$	0	0
$397$	6710.89	0.848388	0.424194	−	0.905571i	$-0.360558\pi$
$397$	6710.89	0.848388	0.424194	+	0.905571i	$0.360558\pi$
$398$	745.190	0.0938518
$399$	0	0
$400$	−15168.2	−1.89602
$401$	6300.52	0.784621	0.392311	−	0.919833i	$-0.371676\pi$
$401$	6300.52	0.784621	0.392311	+	0.919833i	$0.371676\pi$
$402$	0	0
$403$	362.109	0.0447592
$404$	−238.681	−0.0293932
$405$	0	0
$406$	2364.98	0.289093
$407$	3386.02	0.412381
$408$	0	0
$409$	−1919.04	−0.232006	−0.116003	−	0.993249i	$-0.537008\pi$
$409$	−1919.04	−0.232006	−0.116003	+	0.993249i	$0.537008\pi$
$410$	−15828.5	−1.90662
$411$	0	0
$412$	26.8689	0.00321295
$413$	−15102.1	−1.79934
$414$	0	0
$415$	19699.6	2.33016
$416$	−118.739	−0.0139943
$417$	0	0
$418$	−12363.5	−1.44669
$419$	−11720.7	−1.36658	−0.683288	−	0.730149i	$-0.739450\pi$
$419$	−11720.7	−1.36658	−0.683288	+	0.730149i	$0.739450\pi$
$420$	0	0
$421$	11285.6	1.30647	0.653236	−	0.757154i	$-0.273411\pi$
$421$	11285.6	1.30647	0.653236	+	0.757154i	$0.273411\pi$
$422$	6933.18	0.799768
$423$	0	0
$424$	16096.8	1.84371
$425$	9526.35	1.08728
$426$	0	0
$427$	5691.75	0.645065
$428$	494.183	0.0558114
$429$	0	0
$430$	16656.5	1.86801
$431$	12144.7	1.35728	0.678641	−	0.734471i	$-0.262569\pi$
$431$	12144.7	1.35728	0.678641	+	0.734471i	$0.262569\pi$
$432$	0	0
$433$	−11643.9	−1.29231	−0.646154	−	0.763207i	$-0.723624\pi$
$433$	−11643.9	−1.29231	−0.646154	+	0.763207i	$0.723624\pi$
$434$	−2685.40	−0.297012
$435$	0	0
$436$	−466.465	−0.0512377
$437$	−21833.0	−2.38996
$438$	0	0
$439$	−4286.79	−0.466054	−0.233027	−	0.972470i	$-0.574863\pi$
$439$	−4286.79	−0.466054	−0.233027	+	0.972470i	$0.574863\pi$
$440$	14240.3	1.54291
$441$	0	0
$442$	−1118.95	−0.120414
$443$	−861.239	−0.0923673	−0.0461836	−	0.998933i	$-0.514706\pi$
$443$	−861.239	−0.0923673	−0.0461836	+	0.998933i	$0.514706\pi$
$444$	0	0
$445$	20115.0	2.14280
$446$	−15226.1	−1.61654
$447$	0	0
$448$	14526.1	1.53191
$449$	−4202.34	−0.441694	−0.220847	−	0.975308i	$-0.570882\pi$
$449$	−4202.34	−0.441694	−0.220847	+	0.975308i	$0.570882\pi$
$450$	0	0
$451$	9527.66	0.994767
$452$	367.232	0.0382149
$453$	0	0
$454$	−11627.8	−1.20202
$455$	5480.76	0.564708
$456$	0	0
$457$	16451.8	1.68399	0.841993	−	0.539488i	$-0.181382\pi$
$457$	16451.8	1.68399	0.841993	+	0.539488i	$0.181382\pi$
$458$	−8486.11	−0.865786
$459$	0	0
$460$	773.355	0.0783867
$461$	3581.86	0.361874	0.180937	−	0.983495i	$-0.442087\pi$
$461$	3581.86	0.361874	0.180937	+	0.983495i	$0.442087\pi$
$462$	0	0
$463$	−5277.24	−0.529706	−0.264853	−	0.964289i	$-0.585323\pi$
$463$	−5277.24	−0.529706	−0.264853	+	0.964289i	$0.585323\pi$
$464$	1909.12	0.191010
$465$	0	0
$466$	−740.091	−0.0735709
$467$	−14263.7	−1.41337	−0.706687	−	0.707526i	$-0.749811\pi$
$467$	−14263.7	−1.41337	−0.706687	+	0.707526i	$0.749811\pi$
$468$	0	0
$469$	9723.98	0.957381
$470$	6559.31	0.643741
$471$	0	0
$472$	−12591.0	−1.22786
$473$	−10026.0	−0.974626
$474$	0	0
$475$	33775.4	3.26258
$476$	−271.918	−0.0261834
$477$	0	0
$478$	665.183	0.0636502
$479$	13713.0	1.30807	0.654034	−	0.756465i	$-0.273075\pi$
$479$	13713.0	1.30807	0.654034	+	0.756465i	$0.273075\pi$
$480$	0	0
$481$	1086.51	0.102995
$482$	19172.4	1.81178
$483$	0	0
$484$	74.2426	0.00697245
$485$	−34791.4	−3.25731
$486$	0	0
$487$	−11160.8	−1.03848	−0.519242	−	0.854627i	$-0.673786\pi$
$487$	−11160.8	−1.03848	−0.519242	+	0.854627i	$0.673786\pi$
$488$	4745.35	0.440189
$489$	0	0
$490$	−22281.9	−2.05427
$491$	6933.48	0.637279	0.318639	−	0.947876i	$-0.396774\pi$
$491$	6933.48	0.637279	0.318639	+	0.947876i	$0.396774\pi$
$492$	0	0
$493$	−1199.02	−0.109536
$494$	−3967.21	−0.361323
$495$	0	0
$496$	−2167.78	−0.196242
$497$	20923.3	1.88841
$498$	0	0
$499$	134.621	0.0120771	0.00603855	−	0.999982i	$-0.498078\pi$
$499$	134.621	0.0120771	0.00603855	+	0.999982i	$0.498078\pi$
$500$	−586.039	−0.0524170
$501$	0	0
$502$	4583.50	0.407513
$503$	−4654.45	−0.412588	−0.206294	−	0.978490i	$-0.566140\pi$
$503$	−4654.45	−0.412588	−0.206294	+	0.978490i	$0.566140\pi$
$504$	0	0
$505$	18088.0	1.59387
$506$	14205.9	1.24808
$507$	0	0
$508$	492.252	0.0429925
$509$	−645.594	−0.0562190	−0.0281095	−	0.999605i	$-0.508949\pi$
$509$	−645.594	−0.0562190	−0.0281095	+	0.999605i	$0.508949\pi$
$510$	0	0
$511$	−16189.6	−1.40154
$512$	12087.5	1.04335
$513$	0	0
$514$	16064.3	1.37854
$515$	−2036.20	−0.174225
$516$	0	0
$517$	−3948.25	−0.335869
$518$	−8057.57	−0.683454
$519$	0	0
$520$	4569.45	0.385353
$521$	−7410.59	−0.623155	−0.311578	−	0.950221i	$-0.600857\pi$
$521$	−7410.59	−0.623155	−0.311578	+	0.950221i	$0.600857\pi$
$522$	0	0
$523$	17542.7	1.46671	0.733356	−	0.679845i	$-0.237953\pi$
$523$	17542.7	1.46671	0.733356	+	0.679845i	$0.237953\pi$
$524$	127.175	0.0106024
$525$	0	0
$526$	−4609.89	−0.382131
$527$	1361.47	0.112536
$528$	0	0
$529$	12919.6	1.06185
$530$	−37514.6	−3.07458
$531$	0	0
$532$	−964.077	−0.0785678
$533$	3057.25	0.248450
$534$	0	0
$535$	−37450.7	−3.02642
$536$	8107.13	0.653311
$537$	0	0
$538$	−6878.52	−0.551216
$539$	13412.2	1.07181
$540$	0	0
$541$	9430.61	0.749452	0.374726	−	0.927136i	$-0.377737\pi$
$541$	9430.61	0.749452	0.374726	+	0.927136i	$0.377737\pi$
$542$	−123.127	−0.00975789
$543$	0	0
$544$	−446.437	−0.0351854
$545$	35350.1	2.77841
$546$	0	0
$547$	2020.15	0.157908	0.0789539	−	0.996878i	$-0.474842\pi$
$547$	2020.15	0.157908	0.0789539	+	0.996878i	$0.474842\pi$
$548$	622.124	0.0484960
$549$	0	0
$550$	−21976.5	−1.70378
$551$	−4251.09	−0.328680
$552$	0	0
$553$	−4116.44	−0.316544
$554$	8140.46	0.624287
$555$	0	0
$556$	−65.2681	−0.00497839
$557$	−389.740	−0.0296478	−0.0148239	−	0.999890i	$-0.504719\pi$
$557$	−389.740	−0.0296478	−0.0148239	+	0.999890i	$0.504719\pi$
$558$	0	0
$559$	−3217.17	−0.243420
$560$	−32810.7	−2.47591
$561$	0	0
$562$	−19709.8	−1.47937
$563$	−1680.19	−0.125775	−0.0628877	−	0.998021i	$-0.520031\pi$
$563$	−1680.19	−0.125775	−0.0628877	+	0.998021i	$0.520031\pi$
$564$	0	0
$565$	−27829.9	−2.07224
$566$	−1318.59	−0.0979229
$567$	0	0
$568$	17444.3	1.28864
$569$	−17076.6	−1.25815	−0.629075	−	0.777345i	$-0.716566\pi$
$569$	−17076.6	−1.25815	−0.629075	+	0.777345i	$0.716566\pi$
$570$	0	0
$571$	−16649.1	−1.22022	−0.610109	−	0.792317i	$-0.708875\pi$
$571$	−16649.1	−1.22022	−0.610109	+	0.792317i	$0.708875\pi$
$572$	−84.5861	−0.00618308
$573$	0	0
$574$	−22672.5	−1.64866
$575$	−38808.7	−2.81467
$576$	0	0
$577$	18039.8	1.30157	0.650784	−	0.759263i	$-0.274440\pi$
$577$	18039.8	1.30157	0.650784	+	0.759263i	$0.274440\pi$
$578$	9466.77	0.681256
$579$	0	0
$580$	150.580	0.0107801
$581$	28217.6	2.01491
$582$	0	0
$583$	22581.2	1.60415
$584$	−13497.7	−0.956400
$585$	0	0
$586$	−27290.9	−1.92385
$587$	9120.89	0.641328	0.320664	−	0.947193i	$-0.396094\pi$
$587$	9120.89	0.641328	0.320664	+	0.947193i	$0.396094\pi$
$588$	0	0
$589$	4827.05	0.337683
$590$	29344.0	2.04758
$591$	0	0
$592$	−6504.44	−0.451572
$593$	−3798.44	−0.263041	−0.131520	−	0.991313i	$-0.541986\pi$
$593$	−3798.44	−0.263041	−0.131520	+	0.991313i	$0.541986\pi$
$594$	0	0
$595$	20606.7	1.41982
$596$	−555.317	−0.0381655
$597$	0	0
$598$	4558.41	0.311718
$599$	3224.19	0.219928	0.109964	−	0.993936i	$-0.464926\pi$
$599$	3224.19	0.219928	0.109964	+	0.993936i	$0.464926\pi$
$600$	0	0
$601$	6429.69	0.436393	0.218197	−	0.975905i	$-0.429983\pi$
$601$	6429.69	0.436393	0.218197	+	0.975905i	$0.429983\pi$
$602$	23858.5	1.61528
$603$	0	0
$604$	317.431	0.0213842
$605$	−5626.32	−0.378087
$606$	0	0
$607$	−3889.56	−0.260086	−0.130043	−	0.991508i	$-0.541512\pi$
$607$	−3889.56	−0.260086	−0.130043	+	0.991508i	$0.541512\pi$
$608$	−1582.83	−0.105580
$609$	0	0
$610$	−11059.3	−0.734063
$611$	−1266.92	−0.0838857
$612$	0	0
$613$	18334.8	1.20805	0.604025	−	0.796966i	$-0.293563\pi$
$613$	18334.8	1.20805	0.604025	+	0.796966i	$0.293563\pi$
$614$	3852.35	0.253205
$615$	0	0
$616$	20397.7	1.33416
$617$	4169.61	0.272062	0.136031	−	0.990705i	$-0.456565\pi$
$617$	4169.61	0.272062	0.136031	+	0.990705i	$0.456565\pi$
$618$	0	0
$619$	−5923.21	−0.384610	−0.192305	−	0.981335i	$-0.561596\pi$
$619$	−5923.21	−0.384610	−0.192305	+	0.981335i	$0.561596\pi$
$620$	−170.981	−0.0110754
$621$	0	0
$622$	4938.15	0.318331
$623$	28812.6	1.85289
$624$	0	0
$625$	13783.8	0.882161
$626$	−10298.1	−0.657498
$627$	0	0
$628$	333.239	0.0211746
$629$	4085.10	0.258956
$630$	0	0
$631$	−28967.5	−1.82754	−0.913771	−	0.406229i	$-0.866843\pi$
$631$	−28967.5	−1.82754	−0.913771	+	0.406229i	$0.866843\pi$
$632$	−3431.98	−0.216007
$633$	0	0
$634$	9662.14	0.605256
$635$	−37304.3	−2.33130
$636$	0	0
$637$	4303.72	0.267692
$638$	2766.03	0.171643
$639$	0	0
$640$	−26457.8	−1.63412
$641$	−22652.4	−1.39581	−0.697907	−	0.716188i	$-0.745885\pi$
$641$	−22652.4	−1.39581	−0.697907	+	0.716188i	$0.745885\pi$
$642$	0	0
$643$	−11734.9	−0.719719	−0.359859	−	0.933007i	$-0.617175\pi$
$643$	−11734.9	−0.719719	−0.359859	+	0.933007i	$0.617175\pi$
$644$	1107.75	0.0677815
$645$	0	0
$646$	−14916.0	−0.908458
$647$	10592.2	0.643621	0.321810	−	0.946804i	$-0.395709\pi$
$647$	10592.2	0.643621	0.321810	+	0.946804i	$0.395709\pi$
$648$	0	0
$649$	−17663.1	−1.06832
$650$	−7051.83	−0.425532
$651$	0	0
$652$	448.295	0.0269273
$653$	19626.6	1.17618	0.588091	−	0.808795i	$-0.299880\pi$
$653$	19626.6	1.17618	0.588091	+	0.808795i	$0.299880\pi$
$654$	0	0
$655$	−9637.71	−0.574926
$656$	−18302.3	−1.08931
$657$	0	0
$658$	9395.48	0.556647
$659$	5563.31	0.328856	0.164428	−	0.986389i	$-0.447422\pi$
$659$	5563.31	0.328856	0.164428	+	0.986389i	$0.447422\pi$
$660$	0	0
$661$	−1537.13	−0.0904497	−0.0452249	−	0.998977i	$-0.514400\pi$
$661$	−1537.13	−0.0904497	−0.0452249	+	0.998977i	$0.514400\pi$
$662$	32755.7	1.92309
$663$	0	0
$664$	23525.7	1.37496
$665$	73060.6	4.26040
$666$	0	0
$667$	4884.59	0.283557
$668$	−198.094	−0.0114738
$669$	0	0
$670$	−18894.1	−1.08947
$671$	6656.95	0.382993
$672$	0	0
$673$	14786.3	0.846911	0.423456	−	0.905917i	$-0.360817\pi$
$673$	14786.3	0.846911	0.423456	+	0.905917i	$0.360817\pi$
$674$	30594.8	1.74847
$675$	0	0
$676$	530.523	0.0301845
$677$	−1268.81	−0.0720298	−0.0360149	−	0.999351i	$-0.511466\pi$
$677$	−1268.81	−0.0720298	−0.0360149	+	0.999351i	$0.511466\pi$
$678$	0	0
$679$	−49834.9	−2.81662
$680$	17180.3	0.968876
$681$	0	0
$682$	−3140.79	−0.176345
$683$	13430.8	0.752435	0.376218	−	0.926531i	$-0.377224\pi$
$683$	13430.8	0.752435	0.376218	+	0.926531i	$0.377224\pi$
$684$	0	0
$685$	−47146.4	−2.62974
$686$	−5612.87	−0.312391
$687$	0	0
$688$	19259.7	1.06725
$689$	7245.89	0.400648
$690$	0	0
$691$	−29.2231	−0.00160882	−0.000804412	−	1.00000i	$-0.500256\pi$
$691$	−29.2231	−0.00160882	−0.000804412		1.00000i	$0.500256\pi$
$692$	774.172	0.0425283
$693$	0	0
$694$	−18590.5	−1.01684
$695$	4946.21	0.269957
$696$	0	0
$697$	11494.7	0.624668
$698$	−21900.7	−1.18761
$699$	0	0
$700$	−1713.67	−0.0925297
$701$	−13868.8	−0.747241	−0.373620	−	0.927582i	$-0.621884\pi$
$701$	−13868.8	−0.747241	−0.373620	+	0.927582i	$0.621884\pi$
$702$	0	0
$703$	14483.6	0.777041
$704$	16989.4	0.909537
$705$	0	0
$706$	24230.2	1.29166
$707$	25909.0	1.37823
$708$	0	0
$709$	−12022.0	−0.636808	−0.318404	−	0.947955i	$-0.603147\pi$
$709$	−12022.0	−0.636808	−0.318404	+	0.947955i	$0.603147\pi$
$710$	−40655.0	−2.14895
$711$	0	0
$712$	24021.8	1.26440
$713$	−5546.39	−0.291324
$714$	0	0
$715$	6410.18	0.335283
$716$	495.010	0.0258371
$717$	0	0
$718$	3140.45	0.163232
$719$	25462.8	1.32073	0.660363	−	0.750947i	$-0.270403\pi$
$719$	25462.8	1.32073	0.660363	+	0.750947i	$0.270403\pi$
$720$	0	0
$721$	−2916.63	−0.150653
$722$	−33794.5	−1.74197
$723$	0	0
$724$	659.438	0.0338506
$725$	−7556.43	−0.387088
$726$	0	0
$727$	−16110.0	−0.821850	−0.410925	−	0.911669i	$-0.634794\pi$
$727$	−16110.0	−0.821850	−0.410925	+	0.911669i	$0.634794\pi$
$728$	6545.23	0.333217
$729$	0	0
$730$	31457.1	1.59490
$731$	−12096.0	−0.612021
$732$	0	0
$733$	−11795.4	−0.594372	−0.297186	−	0.954820i	$-0.596048\pi$
$733$	−11795.4	−0.594372	−0.297186	+	0.954820i	$0.596048\pi$
$734$	6173.25	0.310435
$735$	0	0
$736$	1818.71	0.0910849
$737$	11373.0	0.568424
$738$	0	0
$739$	859.661	0.0427918	0.0213959	−	0.999771i	$-0.493189\pi$
$739$	859.661	0.0427918	0.0213959	+	0.999771i	$0.493189\pi$
$740$	−513.030	−0.0254856
$741$	0	0
$742$	−53735.5	−2.65861
$743$	27738.6	1.36962	0.684812	−	0.728720i	$-0.259884\pi$
$743$	27738.6	1.36962	0.684812	+	0.728720i	$0.259884\pi$
$744$	0	0
$745$	42083.5	2.06956
$746$	24737.9	1.21410
$747$	0	0
$748$	−318.029	−0.0155458
$749$	−53643.9	−2.61696
$750$	0	0
$751$	−2287.94	−0.111169	−0.0555847	−	0.998454i	$-0.517702\pi$
$751$	−2287.94	−0.111169	−0.0555847	+	0.998454i	$0.517702\pi$
$752$	7584.46	0.367788
$753$	0	0
$754$	887.567	0.0428691
$755$	−24055.8	−1.15958
$756$	0	0
$757$	23515.8	1.12906	0.564530	−	0.825413i	$-0.309058\pi$
$757$	23515.8	1.12906	0.564530	+	0.825413i	$0.309058\pi$
$758$	−29675.9	−1.42200
$759$	0	0
$760$	60912.5	2.90727
$761$	−5118.88	−0.243836	−0.121918	−	0.992540i	$-0.538904\pi$
$761$	−5118.88	−0.243836	−0.121918	+	0.992540i	$0.538904\pi$
$762$	0	0
$763$	50635.1	2.40251
$764$	−367.350	−0.0173956
$765$	0	0
$766$	−8769.14	−0.413632
$767$	−5667.76	−0.266820
$768$	0	0
$769$	−32713.5	−1.53404	−0.767021	−	0.641622i	$-0.778262\pi$
$769$	−32713.5	−1.53404	−0.767021	+	0.641622i	$0.778262\pi$
$770$	−47537.9	−2.22487
$771$	0	0
$772$	309.687	0.0144376
$773$	−1838.02	−0.0855225	−0.0427612	−	0.999085i	$-0.513615\pi$
$773$	−1838.02	−0.0855225	−0.0427612	+	0.999085i	$0.513615\pi$
$774$	0	0
$775$	8580.22	0.397691
$776$	−41548.6	−1.92205
$777$	0	0
$778$	32641.3	1.50418
$779$	40754.3	1.87442
$780$	0	0
$781$	24471.5	1.12120
$782$	17138.9	0.783739
$783$	0	0
$784$	−25764.4	−1.17367
$785$	−25253.8	−1.14821
$786$	0	0
$787$	31329.4	1.41902	0.709512	−	0.704693i	$-0.248915\pi$
$787$	31329.4	1.41902	0.709512	+	0.704693i	$0.248915\pi$
$788$	−726.536	−0.0328449
$789$	0	0
$790$	7998.41	0.360216
$791$	−39863.3	−1.79188
$792$	0	0
$793$	2136.09	0.0956554
$794$	−18677.7	−0.834820
$795$	0	0
$796$	67.9622	0.00302620
$797$	7083.13	0.314802	0.157401	−	0.987535i	$-0.449688\pi$
$797$	7083.13	0.314802	0.157401	+	0.987535i	$0.449688\pi$
$798$	0	0
$799$	−4763.40	−0.210910
$800$	−2813.53	−0.124342
$801$	0	0
$802$	−17535.6	−0.772073
$803$	−18935.0	−0.832132
$804$	0	0
$805$	−83948.2	−3.67551
$806$	−1007.82	−0.0440434
$807$	0	0
$808$	21601.0	0.940496
$809$	−40383.7	−1.75502	−0.877512	−	0.479555i	$-0.840798\pi$
$809$	−40383.7	−1.75502	−0.877512	+	0.479555i	$0.840798\pi$
$810$	0	0
$811$	−16572.3	−0.717549	−0.358774	−	0.933424i	$-0.616805\pi$
$811$	−16572.3	−0.717549	−0.358774	+	0.933424i	$0.616805\pi$
$812$	215.689	0.00932166
$813$	0	0
$814$	−9423.96	−0.405786
$815$	−33973.1	−1.46015
$816$	0	0
$817$	−42886.1	−1.83647
$818$	5341.06	0.228296
$819$	0	0
$820$	−1443.57	−0.0614778
$821$	2812.98	0.119578	0.0597890	−	0.998211i	$-0.480957\pi$
$821$	2812.98	0.119578	0.0597890	+	0.998211i	$0.480957\pi$
$822$	0	0
$823$	14450.1	0.612029	0.306015	−	0.952027i	$-0.401004\pi$
$823$	14450.1	0.612029	0.306015	+	0.952027i	$0.401004\pi$
$824$	−2431.67	−0.102805
$825$	0	0
$826$	42032.1	1.77056
$827$	−25351.1	−1.06595	−0.532977	−	0.846130i	$-0.678927\pi$
$827$	−25351.1	−1.06595	−0.532977	+	0.846130i	$0.678927\pi$
$828$	0	0
$829$	23430.1	0.981617	0.490808	−	0.871268i	$-0.336702\pi$
$829$	23430.1	0.981617	0.490808	+	0.871268i	$0.336702\pi$
$830$	−54827.9	−2.29290
$831$	0	0
$832$	5451.59	0.227163
$833$	16181.2	0.673045
$834$	0	0
$835$	15012.1	0.622174
$836$	−1127.56	−0.0466479
$837$	0	0
$838$	32621.1	1.34472
$839$	36286.4	1.49314	0.746572	−	0.665305i	$-0.231699\pi$
$839$	36286.4	1.49314	0.746572	+	0.665305i	$0.231699\pi$
$840$	0	0
$841$	−23437.9	−0.961004
$842$	−31409.9	−1.28558
$843$	0	0
$844$	632.314	0.0257881
$845$	−40204.6	−1.63678
$846$	0	0
$847$	−8059.08	−0.326934
$848$	−43377.7	−1.75660
$849$	0	0
$850$	−26513.7	−1.06990
$851$	−16642.0	−0.670364
$852$	0	0
$853$	−32090.8	−1.28812	−0.644060	−	0.764975i	$-0.722751\pi$
$853$	−32090.8	−1.28812	−0.644060	+	0.764975i	$0.722751\pi$
$854$	−15841.2	−0.634749
$855$	0	0
$856$	−44724.3	−1.78580
$857$	−13271.5	−0.528992	−0.264496	−	0.964387i	$-0.585206\pi$
$857$	−13271.5	−0.528992	−0.264496	+	0.964387i	$0.585206\pi$
$858$	0	0
$859$	−31122.1	−1.23617	−0.618086	−	0.786110i	$-0.712092\pi$
$859$	−31122.1	−1.23617	−0.618086	+	0.786110i	$0.712092\pi$
$860$	1519.09	0.0602331
$861$	0	0
$862$	−33801.0	−1.33558
$863$	−29880.8	−1.17863	−0.589313	−	0.807905i	$-0.700601\pi$
$863$	−29880.8	−1.17863	−0.589313	+	0.807905i	$0.700601\pi$
$864$	0	0
$865$	−58669.0	−2.30614
$866$	32407.2	1.27164
$867$	0	0
$868$	−244.912	−0.00957700
$869$	−4814.50	−0.187941
$870$	0	0
$871$	3649.37	0.141968
$872$	42215.7	1.63946
$873$	0	0
$874$	60765.4	2.35174
$875$	63614.9	2.45780
$876$	0	0
$877$	14277.6	0.549737	0.274869	−	0.961482i	$-0.411366\pi$
$877$	14277.6	0.549737	0.274869	+	0.961482i	$0.411366\pi$
$878$	11931.0	0.458600
$879$	0	0
$880$	−38374.8	−1.47001
$881$	29068.3	1.11162	0.555809	−	0.831310i	$-0.312409\pi$
$881$	29068.3	1.11162	0.555809	+	0.831310i	$0.312409\pi$
$882$	0	0
$883$	19579.3	0.746203	0.373102	−	0.927790i	$-0.378294\pi$
$883$	19579.3	0.746203	0.373102	+	0.927790i	$0.378294\pi$
$884$	−102.050	−0.00388269
$885$	0	0
$886$	2397.00	0.0908901
$887$	−1135.24	−0.0429738	−0.0214869	−	0.999769i	$-0.506840\pi$
$887$	−1135.24	−0.0429738	−0.0214869	+	0.999769i	$0.506840\pi$
$888$	0	0
$889$	−53434.3	−2.01589
$890$	−55984.1	−2.10853
$891$	0	0
$892$	−1388.64	−0.0521244
$893$	−16888.5	−0.632870
$894$	0	0
$895$	−37513.3	−1.40104
$896$	−37897.9	−1.41304
$897$	0	0
$898$	11695.9	0.434631
$899$	−1079.94	−0.0400644
$900$	0	0
$901$	27243.3	1.00733
$902$	−26517.3	−0.978859
$903$	0	0
$904$	−33235.0	−1.22277
$905$	−49974.1	−1.83558
$906$	0	0
$907$	−42319.8	−1.54929	−0.774645	−	0.632396i	$-0.782072\pi$
$907$	−42319.8	−1.54929	−0.774645	+	0.632396i	$0.782072\pi$
$908$	−1060.47	−0.0387586
$909$	0	0
$910$	−15254.0	−0.555677
$911$	−20188.8	−0.734233	−0.367116	−	0.930175i	$-0.619655\pi$
$911$	−20188.8	−0.734233	−0.367116	+	0.930175i	$0.619655\pi$
$912$	0	0
$913$	33002.7	1.19631
$914$	−45788.5	−1.65706
$915$	0	0
$916$	−773.942	−0.0279168
$917$	−13804.9	−0.497142
$918$	0	0
$919$	29559.7	1.06103	0.530514	−	0.847676i	$-0.321999\pi$
$919$	29559.7	1.06103	0.530514	+	0.847676i	$0.321999\pi$
$920$	−69989.8	−2.50815
$921$	0	0
$922$	−9969.01	−0.356087
$923$	7852.44	0.280029
$924$	0	0
$925$	25745.0	0.915125
$926$	14687.6	0.521235
$927$	0	0
$928$	354.120	0.0125265
$929$	35279.7	1.24595	0.622975	−	0.782242i	$-0.285924\pi$
$929$	35279.7	1.24595	0.622975	+	0.782242i	$0.285924\pi$
$930$	0	0
$931$	57370.2	2.01958
$932$	−67.4971	−0.00237225
$933$	0	0
$934$	39698.7	1.39077
$935$	24101.2	0.842987
$936$	0	0
$937$	−33500.7	−1.16801	−0.584003	−	0.811751i	$-0.698514\pi$
$937$	−33500.7	−1.16801	−0.584003	+	0.811751i	$0.698514\pi$
$938$	−27063.7	−0.942070
$939$	0	0
$940$	598.216	0.0207571
$941$	51286.9	1.77673	0.888366	−	0.459137i	$-0.151841\pi$
$941$	51286.9	1.77673	0.888366	+	0.459137i	$0.151841\pi$
$942$	0	0
$943$	−46827.5	−1.61709
$944$	33930.2	1.16985
$945$	0	0
$946$	27904.4	0.959040
$947$	−23115.3	−0.793186	−0.396593	−	0.917995i	$-0.629808\pi$
$947$	−23115.3	−0.793186	−0.396593	+	0.917995i	$0.629808\pi$
$948$	0	0
$949$	−6075.89	−0.207831
$950$	−94003.6	−3.21040
$951$	0	0
$952$	24608.9	0.837794
$953$	54712.6	1.85972	0.929861	−	0.367911i	$-0.119927\pi$
$953$	54712.6	1.85972	0.929861	+	0.367911i	$0.119927\pi$
$954$	0	0
$955$	27838.9	0.943293
$956$	60.6655	0.00205237
$957$	0	0
$958$	−38166.1	−1.28715
$959$	−67531.9	−2.27395
$960$	0	0
$961$	−28564.7	−0.958838
$962$	−3023.97	−0.101348
$963$	0	0
$964$	1748.54	0.0584199
$965$	−23469.0	−0.782894
$966$	0	0
$967$	−53949.5	−1.79411	−0.897053	−	0.441923i	$-0.854296\pi$
$967$	−53949.5	−1.79411	−0.897053	+	0.441923i	$0.854296\pi$
$968$	−6719.07	−0.223098
$969$	0	0
$970$	96831.3	3.20522
$971$	54124.5	1.78881	0.894407	−	0.447253i	$-0.147598\pi$
$971$	54124.5	1.78881	0.894407	+	0.447253i	$0.147598\pi$
$972$	0	0
$973$	7084.90	0.233434
$974$	31062.6	1.02188
$975$	0	0
$976$	−12787.8	−0.419392
$977$	58614.1	1.91938	0.959688	−	0.281067i	$-0.0906882\pi$
$977$	58614.1	1.91938	0.959688	+	0.281067i	$0.0906882\pi$
$978$	0	0
$979$	33698.6	1.10011
$980$	−2032.14	−0.0662390
$981$	0	0
$982$	−19297.2	−0.627087
$983$	18587.9	0.603116	0.301558	−	0.953448i	$-0.402493\pi$
$983$	18587.9	0.603116	0.301558	+	0.953448i	$0.402493\pi$
$984$	0	0
$985$	55059.1	1.78104
$986$	3337.10	0.107784
$987$	0	0
$988$	−361.814	−0.0116507
$989$	49277.1	1.58435
$990$	0	0
$991$	13247.9	0.424657	0.212328	−	0.977198i	$-0.431895\pi$
$991$	13247.9	0.424657	0.212328	+	0.977198i	$0.431895\pi$
$992$	−402.099	−0.0128696
$993$	0	0
$994$	−58233.7	−1.85821
$995$	−5150.37	−0.164098
$996$	0	0
$997$	33371.1	1.06005	0.530026	−	0.847981i	$-0.322182\pi$
$997$	33371.1	1.06005	0.530026	+	0.847981i	$0.322182\pi$
$998$	−374.677	−0.0118840
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.h.1.18	yes	59
3.2	odd	2		2151.4.a.g.1.42	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.42	✓	59	3.2	odd	2
2151.4.a.h.1.18	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)

\(f(q)\)	\(=\)	\(q-2.78319 q^{2} -0.253830 q^{4} +19.2360 q^{5} +27.5534 q^{7} +22.9720 q^{8} +O(q^{10})\)	\(q-2.78319 q^{2} -0.253830 q^{4} +19.2360 q^{5} +27.5534 q^{7} +22.9720 q^{8} -53.5375 q^{10} +32.2259 q^{11} +10.3407 q^{13} -76.6866 q^{14} -61.9049 q^{16} +38.8792 q^{17} +137.845 q^{19} -4.88268 q^{20} -89.6910 q^{22} -158.387 q^{23} +245.024 q^{25} -28.7802 q^{26} -6.99390 q^{28} -30.8395 q^{29} +35.0179 q^{31} -11.4827 q^{32} -108.208 q^{34} +530.018 q^{35} +105.071 q^{37} -383.651 q^{38} +441.890 q^{40} +295.652 q^{41} -311.117 q^{43} -8.17992 q^{44} +440.823 q^{46} -122.518 q^{47} +416.192 q^{49} -681.949 q^{50} -2.62478 q^{52} +700.716 q^{53} +619.898 q^{55} +632.958 q^{56} +85.8324 q^{58} -548.102 q^{59} +206.571 q^{61} -97.4616 q^{62} +527.198 q^{64} +198.914 q^{65} +352.913 q^{67} -9.86873 q^{68} -1475.14 q^{70} +759.373 q^{71} -587.570 q^{73} -292.434 q^{74} -34.9894 q^{76} +887.935 q^{77} -149.398 q^{79} -1190.80 q^{80} -822.857 q^{82} +1024.10 q^{83} +747.881 q^{85} +865.900 q^{86} +740.295 q^{88} +1045.70 q^{89} +284.922 q^{91} +40.2035 q^{92} +340.991 q^{94} +2651.60 q^{95} -1808.66 q^{97} -1158.34 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) \) 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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Database

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