LMFDB - Embedded newform 2151.4.a.h.1.20

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			1.20
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)

$f(q)$	$=$	$q-1.92673 q^{2} -4.28770 q^{4} -4.54299 q^{5} -14.1149 q^{7} +23.6751 q^{8} +O(q^{10})$	\(q-1.92673 q^{2} -4.28770 q^{4} -4.54299 q^{5} -14.1149 q^{7} +23.6751 q^{8} +8.75313 q^{10} -28.0304 q^{11} -31.9364 q^{13} +27.1957 q^{14} -11.3141 q^{16} -71.9548 q^{17} +110.437 q^{19} +19.4790 q^{20} +54.0072 q^{22} +55.0861 q^{23} -104.361 q^{25} +61.5328 q^{26} +60.5204 q^{28} +12.1306 q^{29} +19.8759 q^{31} -167.602 q^{32} +138.638 q^{34} +64.1239 q^{35} -243.779 q^{37} -212.783 q^{38} -107.556 q^{40} -349.249 q^{41} -243.760 q^{43} +120.186 q^{44} -106.136 q^{46} -96.5623 q^{47} -143.769 q^{49} +201.076 q^{50} +136.933 q^{52} -728.153 q^{53} +127.342 q^{55} -334.172 q^{56} -23.3724 q^{58} +259.682 q^{59} +616.386 q^{61} -38.2956 q^{62} +413.436 q^{64} +145.087 q^{65} -34.5550 q^{67} +308.520 q^{68} -123.550 q^{70} +110.119 q^{71} -908.850 q^{73} +469.697 q^{74} -473.522 q^{76} +395.647 q^{77} +82.5884 q^{79} +51.3996 q^{80} +672.910 q^{82} -616.389 q^{83} +326.890 q^{85} +469.661 q^{86} -663.624 q^{88} +700.181 q^{89} +450.779 q^{91} -236.193 q^{92} +186.050 q^{94} -501.716 q^{95} -1734.66 q^{97} +277.005 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.92673	−0.681203	−0.340602	−	0.940208i	$-0.610631\pi$
$2$	−1.92673	−0.681203	−0.340602	+	0.940208i	$0.610631\pi$
$3$	0	0
$3$	0	0
$4$	−4.28770	−0.535962
$5$	−4.54299	−0.406337	−0.203169	−	0.979144i	$-0.565124\pi$
$5$	−4.54299	−0.406337	−0.203169	+	0.979144i	$0.565124\pi$
$6$	0	0
$7$	−14.1149	−0.762133	−0.381067	−	0.924548i	$-0.624443\pi$
$7$	−14.1149	−0.762133	−0.381067	+	0.924548i	$0.624443\pi$
$8$	23.6751	1.04630
$9$	0	0
$10$	8.75313	0.276798
$11$	−28.0304	−0.768318	−0.384159	−	0.923267i	$-0.625509\pi$
$11$	−28.0304	−0.768318	−0.384159	+	0.923267i	$0.625509\pi$
$12$	0	0
$13$	−31.9364	−0.681350	−0.340675	−	0.940181i	$-0.610656\pi$
$13$	−31.9364	−0.681350	−0.340675	+	0.940181i	$0.610656\pi$
$14$	27.1957	0.519167
$15$	0	0
$16$	−11.3141	−0.176782
$17$	−71.9548	−1.02656	−0.513282	−	0.858220i	$-0.671571\pi$
$17$	−71.9548	−1.02656	−0.513282	+	0.858220i	$0.671571\pi$
$18$	0	0
$19$	110.437	1.33348	0.666739	−	0.745292i	$-0.267690\pi$
$19$	110.437	1.33348	0.666739	+	0.745292i	$0.267690\pi$
$20$	19.4790	0.217781
$21$	0	0
$22$	54.0072	0.523381
$23$	55.0861	0.499402	0.249701	−	0.968323i	$-0.419668\pi$
$23$	55.0861	0.499402	0.249701	+	0.968323i	$0.419668\pi$
$24$	0	0
$25$	−104.361	−0.834890
$26$	61.5328	0.464138
$27$	0	0
$28$	60.5204	0.408475
$29$	12.1306	0.0776755	0.0388378	−	0.999246i	$-0.487634\pi$
$29$	12.1306	0.0776755	0.0388378	+	0.999246i	$0.487634\pi$
$30$	0	0
$31$	19.8759	0.115156	0.0575778	−	0.998341i	$-0.481662\pi$
$31$	19.8759	0.115156	0.0575778	+	0.998341i	$0.481662\pi$
$32$	−167.602	−0.925878
$33$	0	0
$34$	138.638	0.699299
$35$	64.1239	0.309683
$36$	0	0
$37$	−243.779	−1.08316	−0.541581	−	0.840649i	$-0.682174\pi$
$37$	−243.779	−1.08316	−0.541581	+	0.840649i	$0.682174\pi$
$38$	−212.783	−0.908369
$39$	0	0
$40$	−107.556	−0.425152
$41$	−349.249	−1.33033	−0.665165	−	0.746697i	$-0.731639\pi$
$41$	−349.249	−1.33033	−0.665165	+	0.746697i	$0.731639\pi$
$42$	0	0
$43$	−243.760	−0.864490	−0.432245	−	0.901756i	$-0.642278\pi$
$43$	−243.760	−0.864490	−0.432245	+	0.901756i	$0.642278\pi$
$44$	120.186	0.411789
$45$	0	0
$46$	−106.136	−0.340194
$47$	−96.5623	−0.299682	−0.149841	−	0.988710i	$-0.547876\pi$
$47$	−96.5623	−0.299682	−0.149841	+	0.988710i	$0.547876\pi$
$48$	0	0
$49$	−143.769	−0.419153
$50$	201.076	0.568730
$51$	0	0
$52$	136.933	0.365178
$53$	−728.153	−1.88716	−0.943580	−	0.331145i	$-0.892565\pi$
$53$	−728.153	−1.88716	−0.943580	+	0.331145i	$0.892565\pi$
$54$	0	0
$55$	127.342	0.312196
$56$	−334.172	−0.797422
$57$	0	0
$58$	−23.3724	−0.0529128
$59$	259.682	0.573013	0.286506	−	0.958078i	$-0.407506\pi$
$59$	259.682	0.573013	0.286506	+	0.958078i	$0.407506\pi$
$60$	0	0
$61$	616.386	1.29377	0.646886	−	0.762586i	$-0.276071\pi$
$61$	616.386	1.29377	0.646886	+	0.762586i	$0.276071\pi$
$62$	−38.2956	−0.0784444
$63$	0	0
$64$	413.436	0.807493
$65$	145.087	0.276858
$66$	0	0
$67$	−34.5550	−0.0630084	−0.0315042	−	0.999504i	$-0.510030\pi$
$67$	−34.5550	−0.0630084	−0.0315042	+	0.999504i	$0.510030\pi$
$68$	308.520	0.550200
$69$	0	0
$70$	−123.550	−0.210957
$71$	110.119	0.184067	0.0920335	−	0.995756i	$-0.470663\pi$
$71$	110.119	0.184067	0.0920335	+	0.995756i	$0.470663\pi$
$72$	0	0
$73$	−908.850	−1.45716	−0.728581	−	0.684959i	$-0.759820\pi$
$73$	−908.850	−1.45716	−0.728581	+	0.684959i	$0.759820\pi$
$74$	469.697	0.737853
$75$	0	0
$76$	−473.522	−0.714694
$77$	395.647	0.585561
$78$	0	0
$79$	82.5884	0.117619	0.0588097	−	0.998269i	$-0.481269\pi$
$79$	82.5884	0.117619	0.0588097	+	0.998269i	$0.481269\pi$
$80$	51.3996	0.0718332
$81$	0	0
$82$	672.910	0.906225
$83$	−616.389	−0.815150	−0.407575	−	0.913172i	$-0.633625\pi$
$83$	−616.389	−0.815150	−0.407575	+	0.913172i	$0.633625\pi$
$84$	0	0
$85$	326.890	0.417131
$86$	469.661	0.588893
$87$	0	0
$88$	−663.624	−0.803893
$89$	700.181	0.833922	0.416961	−	0.908924i	$-0.363095\pi$
$89$	700.181	0.833922	0.416961	+	0.908924i	$0.363095\pi$
$90$	0	0
$91$	450.779	0.519279
$92$	−236.193	−0.267661
$93$	0	0
$94$	186.050	0.204144
$95$	−501.716	−0.541842
$96$	0	0
$97$	−1734.66	−1.81576	−0.907879	−	0.419233i	$-0.862299\pi$
$97$	−1734.66	−1.81576	−0.907879	+	0.419233i	$0.862299\pi$
$98$	277.005	0.285528
$99$	0	0
$100$	447.470	0.447470
$101$	124.059	0.122221	0.0611104	−	0.998131i	$-0.480536\pi$
$101$	124.059	0.122221	0.0611104	+	0.998131i	$0.480536\pi$
$102$	0	0
$103$	−1711.43	−1.63721	−0.818604	−	0.574358i	$-0.805252\pi$
$103$	−1711.43	−1.63721	−0.818604	+	0.574358i	$0.805252\pi$
$104$	−756.097	−0.712898
$105$	0	0
$106$	1402.96	1.28554
$107$	1312.09	1.18547	0.592733	−	0.805399i	$-0.298049\pi$
$107$	1312.09	1.18547	0.592733	+	0.805399i	$0.298049\pi$
$108$	0	0
$109$	−1547.57	−1.35991	−0.679955	−	0.733254i	$-0.738001\pi$
$109$	−1547.57	−1.35991	−0.679955	+	0.733254i	$0.738001\pi$
$110$	−245.354	−0.212669
$111$	0	0
$112$	159.697	0.134732
$113$	−1304.61	−1.08608	−0.543040	−	0.839707i	$-0.682727\pi$
$113$	−1304.61	−1.08608	−0.543040	+	0.839707i	$0.682727\pi$
$114$	0	0
$115$	−250.256	−0.202926
$116$	−52.0122	−0.0416312
$117$	0	0
$118$	−500.339	−0.390338
$119$	1015.63	0.782379
$120$	0	0
$121$	−545.294	−0.409688
$122$	−1187.61	−0.881322
$123$	0	0
$124$	−85.2221	−0.0617191
$125$	1041.99	0.745584
$126$	0	0
$127$	479.592	0.335094	0.167547	−	0.985864i	$-0.446415\pi$
$127$	479.592	0.335094	0.167547	+	0.985864i	$0.446415\pi$
$128$	544.233	0.375811
$129$	0	0
$130$	−279.543	−0.188596
$131$	1823.07	1.21590	0.607949	−	0.793976i	$-0.291992\pi$
$131$	1823.07	1.21590	0.607949	+	0.793976i	$0.291992\pi$
$132$	0	0
$133$	−1558.81	−1.01629
$134$	66.5782	0.0429215
$135$	0	0
$136$	−1703.54	−1.07410
$137$	−1887.11	−1.17684	−0.588419	−	0.808556i	$-0.700249\pi$
$137$	−1887.11	−1.17684	−0.588419	+	0.808556i	$0.700249\pi$
$138$	0	0
$139$	465.291	0.283924	0.141962	−	0.989872i	$-0.454659\pi$
$139$	465.291	0.283924	0.141962	+	0.989872i	$0.454659\pi$
$140$	−274.944	−0.165978
$141$	0	0
$142$	−212.170	−0.125387
$143$	895.190	0.523493
$144$	0	0
$145$	−55.1091	−0.0315625
$146$	1751.11	0.992624
$147$	0	0
$148$	1045.25	0.580534
$149$	2487.89	1.36789	0.683947	−	0.729532i	$-0.260262\pi$
$149$	2487.89	1.36789	0.683947	+	0.729532i	$0.260262\pi$
$150$	0	0
$151$	−3323.76	−1.79129	−0.895643	−	0.444774i	$-0.853284\pi$
$151$	−3323.76	−1.79129	−0.895643	+	0.444774i	$0.853284\pi$
$152$	2614.62	1.39522
$153$	0	0
$154$	−762.306	−0.398886
$155$	−90.2962	−0.0467920
$156$	0	0
$157$	−720.418	−0.366214	−0.183107	−	0.983093i	$-0.558615\pi$
$157$	−720.418	−0.366214	−0.183107	+	0.983093i	$0.558615\pi$
$158$	−159.126	−0.0801226
$159$	0	0
$160$	761.413	0.376219
$161$	−777.535	−0.380611
$162$	0	0
$163$	1108.25	0.532545	0.266272	−	0.963898i	$-0.414208\pi$
$163$	1108.25	0.532545	0.266272	+	0.963898i	$0.414208\pi$
$164$	1497.47	0.713007
$165$	0	0
$166$	1187.62	0.555283
$167$	−1676.79	−0.776968	−0.388484	−	0.921456i	$-0.627001\pi$
$167$	−1676.79	−0.776968	−0.388484	+	0.921456i	$0.627001\pi$
$168$	0	0
$169$	−1177.07	−0.535762
$170$	−629.829	−0.284151
$171$	0	0
$172$	1045.17	0.463334
$173$	−3900.26	−1.71405	−0.857026	−	0.515274i	$-0.827690\pi$
$173$	−3900.26	−1.71405	−0.857026	+	0.515274i	$0.827690\pi$
$174$	0	0
$175$	1473.05	0.636297
$176$	317.138	0.135825
$177$	0	0
$178$	−1349.06	−0.568070
$179$	663.778	0.277168	0.138584	−	0.990351i	$-0.455745\pi$
$179$	663.778	0.277168	0.138584	+	0.990351i	$0.455745\pi$
$180$	0	0
$181$	2098.72	0.861859	0.430929	−	0.902386i	$-0.358186\pi$
$181$	2098.72	0.861859	0.430929	+	0.902386i	$0.358186\pi$
$182$	−868.530	−0.353735
$183$	0	0
$184$	1304.17	0.522526
$185$	1107.48	0.440129
$186$	0	0
$187$	2016.92	0.788728
$188$	414.030	0.160618
$189$	0	0
$190$	966.673	0.369104
$191$	302.401	0.114560	0.0572801	−	0.998358i	$-0.481757\pi$
$191$	302.401	0.114560	0.0572801	+	0.998358i	$0.481757\pi$
$192$	0	0
$193$	−3757.87	−1.40154	−0.700770	−	0.713387i	$-0.747160\pi$
$193$	−3757.87	−1.40154	−0.700770	+	0.713387i	$0.747160\pi$
$194$	3342.24	1.23690
$195$	0	0
$196$	616.440	0.224650
$197$	2992.66	1.08233	0.541163	−	0.840918i	$-0.317984\pi$
$197$	2992.66	1.08233	0.541163	+	0.840918i	$0.317984\pi$
$198$	0	0
$199$	−574.375	−0.204605	−0.102302	−	0.994753i	$-0.532621\pi$
$199$	−574.375	−0.204605	−0.102302	+	0.994753i	$0.532621\pi$
$200$	−2470.76	−0.873547
$201$	0	0
$202$	−239.028	−0.0832572
$203$	−171.222	−0.0591991
$204$	0	0
$205$	1586.63	0.540563
$206$	3297.47	1.11527
$207$	0	0
$208$	361.330	0.120451
$209$	−3095.61	−1.02453
$210$	0	0
$211$	−1130.97	−0.369002	−0.184501	−	0.982832i	$-0.559067\pi$
$211$	−1130.97	−0.369002	−0.184501	+	0.982832i	$0.559067\pi$
$212$	3122.10	1.01145
$213$	0	0
$214$	−2528.05	−0.807543
$215$	1107.40	0.351274
$216$	0	0
$217$	−280.547	−0.0877639
$218$	2981.75	0.926375
$219$	0	0
$220$	−546.004	−0.167325
$221$	2297.97	0.699450
$222$	0	0
$223$	4758.26	1.42886	0.714432	−	0.699705i	$-0.246685\pi$
$223$	4758.26	1.42886	0.714432	+	0.699705i	$0.246685\pi$
$224$	2365.68	0.705642
$225$	0	0
$226$	2513.63	0.739841
$227$	4921.32	1.43894	0.719470	−	0.694523i	$-0.244385\pi$
$227$	4921.32	1.43894	0.719470	+	0.694523i	$0.244385\pi$
$228$	0	0
$229$	−5415.04	−1.56260	−0.781301	−	0.624154i	$-0.785444\pi$
$229$	−5415.04	−1.56260	−0.781301	+	0.624154i	$0.785444\pi$
$230$	482.176	0.138234
$231$	0	0
$232$	287.193	0.0812721
$233$	4829.08	1.35778	0.678892	−	0.734238i	$-0.262460\pi$
$233$	4829.08	1.35778	0.678892	+	0.734238i	$0.262460\pi$
$234$	0	0
$235$	438.682	0.121772
$236$	−1113.44	−0.307113
$237$	0	0
$238$	−1956.86	−0.532959
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	5048.84	1.34948	0.674739	−	0.738056i	$-0.264256\pi$
$241$	5048.84	1.34948	0.674739	+	0.738056i	$0.264256\pi$
$242$	1050.64	0.279080
$243$	0	0
$244$	−2642.88	−0.693413
$245$	653.143	0.170318
$246$	0	0
$247$	−3526.97	−0.908565
$248$	470.565	0.120488
$249$	0	0
$250$	−2007.63	−0.507894
$251$	6977.00	1.75452	0.877259	−	0.480017i	$-0.159370\pi$
$251$	6977.00	1.75452	0.877259	+	0.480017i	$0.159370\pi$
$252$	0	0
$253$	−1544.09	−0.383700
$254$	−924.046	−0.228267
$255$	0	0
$256$	−4356.08	−1.06350
$257$	−7599.68	−1.84457	−0.922286	−	0.386507i	$-0.873681\pi$
$257$	−7599.68	−1.84457	−0.922286	+	0.386507i	$0.873681\pi$
$258$	0	0
$259$	3440.91	0.825514
$260$	−622.087	−0.148385
$261$	0	0
$262$	−3512.58	−0.828274
$263$	1590.80	0.372977	0.186489	−	0.982457i	$-0.440289\pi$
$263$	1590.80	0.372977	0.186489	+	0.982457i	$0.440289\pi$
$264$	0	0
$265$	3307.99	0.766823
$266$	3003.42	0.692298
$267$	0	0
$268$	148.161	0.0337701
$269$	−4255.95	−0.964645	−0.482323	−	0.875994i	$-0.660207\pi$
$269$	−4255.95	−0.964645	−0.482323	+	0.875994i	$0.660207\pi$
$270$	0	0
$271$	8311.36	1.86302	0.931511	−	0.363713i	$-0.118491\pi$
$271$	8311.36	1.86302	0.931511	+	0.363713i	$0.118491\pi$
$272$	814.100	0.181478
$273$	0	0
$274$	3635.96	0.801665
$275$	2925.29	0.641461
$276$	0	0
$277$	3888.34	0.843421	0.421710	−	0.906731i	$-0.361430\pi$
$277$	3888.34	0.843421	0.421710	+	0.906731i	$0.361430\pi$
$278$	−896.492	−0.193410
$279$	0	0
$280$	1518.14	0.324022
$281$	2351.52	0.499217	0.249609	−	0.968347i	$-0.419698\pi$
$281$	2351.52	0.499217	0.249609	+	0.968347i	$0.419698\pi$
$282$	0	0
$283$	−1830.05	−0.384400	−0.192200	−	0.981356i	$-0.561562\pi$
$283$	−1830.05	−0.384400	−0.192200	+	0.981356i	$0.561562\pi$
$284$	−472.158	−0.0986529
$285$	0	0
$286$	−1724.79	−0.356605
$287$	4929.61	1.01389
$288$	0	0
$289$	264.488	0.0538344
$290$	106.180	0.0215005
$291$	0	0
$292$	3896.88	0.780984
$293$	4939.16	0.984808	0.492404	−	0.870367i	$-0.336118\pi$
$293$	4939.16	0.984808	0.492404	+	0.870367i	$0.336118\pi$
$294$	0	0
$295$	−1179.73	−0.232837
$296$	−5771.49	−1.13331
$297$	0	0
$298$	−4793.51	−0.931814
$299$	−1759.25	−0.340268
$300$	0	0
$301$	3440.65	0.658856
$302$	6404.01	1.22023
$303$	0	0
$304$	−1249.49	−0.235735
$305$	−2800.24	−0.525708
$306$	0	0
$307$	−5082.19	−0.944808	−0.472404	−	0.881382i	$-0.656614\pi$
$307$	−5082.19	−0.944808	−0.472404	+	0.881382i	$0.656614\pi$
$308$	−1696.42	−0.313838
$309$	0	0
$310$	173.977	0.0318749
$311$	−249.779	−0.0455424	−0.0227712	−	0.999741i	$-0.507249\pi$
$311$	−249.779	−0.0455424	−0.0227712	+	0.999741i	$0.507249\pi$
$312$	0	0
$313$	1335.72	0.241213	0.120606	−	0.992700i	$-0.461516\pi$
$313$	1335.72	0.241213	0.120606	+	0.992700i	$0.461516\pi$
$314$	1388.05	0.249466
$315$	0	0
$316$	−354.114	−0.0630395
$317$	4099.50	0.726344	0.363172	−	0.931722i	$-0.381694\pi$
$317$	4099.50	0.726344	0.363172	+	0.931722i	$0.381694\pi$
$318$	0	0
$319$	−340.025	−0.0596795
$320$	−1878.24	−0.328115
$321$	0	0
$322$	1498.10	0.259273
$323$	−7946.50	−1.36890
$324$	0	0
$325$	3332.92	0.568852
$326$	−2135.30	−0.362771
$327$	0	0
$328$	−8268.51	−1.39193
$329$	1362.97	0.228398
$330$	0	0
$331$	−704.646	−0.117012	−0.0585058	−	0.998287i	$-0.518634\pi$
$331$	−704.646	−0.117012	−0.0585058	+	0.998287i	$0.518634\pi$
$332$	2642.89	0.436890
$333$	0	0
$334$	3230.72	0.529273
$335$	156.983	0.0256027
$336$	0	0
$337$	−3937.55	−0.636474	−0.318237	−	0.948011i	$-0.603091\pi$
$337$	−3937.55	−0.636474	−0.318237	+	0.948011i	$0.603091\pi$
$338$	2267.90	0.364963
$339$	0	0
$340$	−1401.60	−0.223567
$341$	−557.132	−0.0884761
$342$	0	0
$343$	6870.70	1.08158
$344$	−5771.05	−0.904518
$345$	0	0
$346$	7514.75	1.16762
$347$	4188.64	0.648006	0.324003	−	0.946056i	$-0.394971\pi$
$347$	4188.64	0.648006	0.324003	+	0.946056i	$0.394971\pi$
$348$	0	0
$349$	−5622.54	−0.862372	−0.431186	−	0.902263i	$-0.641905\pi$
$349$	−5622.54	−0.862372	−0.431186	+	0.902263i	$0.641905\pi$
$350$	−2838.17	−0.433448
$351$	0	0
$352$	4697.95	0.711368
$353$	−10770.7	−1.62399	−0.811994	−	0.583665i	$-0.801618\pi$
$353$	−10770.7	−1.62399	−0.811994	+	0.583665i	$0.801618\pi$
$354$	0	0
$355$	−500.271	−0.0747933
$356$	−3002.17	−0.446951
$357$	0	0
$358$	−1278.92	−0.188808
$359$	4653.73	0.684163	0.342081	−	0.939670i	$-0.388868\pi$
$359$	4653.73	0.684163	0.342081	+	0.939670i	$0.388868\pi$
$360$	0	0
$361$	5337.41	0.778162
$362$	−4043.67	−0.587101
$363$	0	0
$364$	−1932.80	−0.278314
$365$	4128.90	0.592099
$366$	0	0
$367$	−12201.2	−1.73541	−0.867706	−	0.497077i	$-0.834407\pi$
$367$	−12201.2	−1.73541	−0.867706	+	0.497077i	$0.834407\pi$
$368$	−623.247	−0.0882854
$369$	0	0
$370$	−2133.83	−0.299817
$371$	10277.8	1.43827
$372$	0	0
$373$	5045.08	0.700333	0.350167	−	0.936687i	$-0.386125\pi$
$373$	5045.08	0.700333	0.350167	+	0.936687i	$0.386125\pi$
$374$	−3886.07	−0.537284
$375$	0	0
$376$	−2286.12	−0.313558
$377$	−387.406	−0.0529242
$378$	0	0
$379$	−4999.72	−0.677621	−0.338810	−	0.940855i	$-0.610024\pi$
$379$	−4999.72	−0.677621	−0.338810	+	0.940855i	$0.610024\pi$
$380$	2151.21	0.290407
$381$	0	0
$382$	−582.647	−0.0780387
$383$	10546.4	1.40704	0.703521	−	0.710674i	$-0.251610\pi$
$383$	10546.4	1.40704	0.703521	+	0.710674i	$0.251610\pi$
$384$	0	0
$385$	−1797.42	−0.237935
$386$	7240.41	0.954734
$387$	0	0
$388$	7437.72	0.973177
$389$	11923.9	1.55415	0.777075	−	0.629408i	$-0.216702\pi$
$389$	11923.9	1.55415	0.777075	+	0.629408i	$0.216702\pi$
$390$	0	0
$391$	−3963.71	−0.512668
$392$	−3403.76	−0.438561
$393$	0	0
$394$	−5766.06	−0.737284
$395$	−375.198	−0.0477931
$396$	0	0
$397$	4067.23	0.514178	0.257089	−	0.966388i	$-0.417237\pi$
$397$	4067.23	0.514178	0.257089	+	0.966388i	$0.417237\pi$
$398$	1106.67	0.139377
$399$	0	0
$400$	1180.75	0.147594
$401$	14266.9	1.77670	0.888350	−	0.459168i	$-0.151852\pi$
$401$	14266.9	1.77670	0.888350	+	0.459168i	$0.151852\pi$
$402$	0	0
$403$	−634.765	−0.0784613
$404$	−531.926	−0.0655057
$405$	0	0
$406$	329.899	0.0403266
$407$	6833.23	0.832213
$408$	0	0
$409$	8198.77	0.991205	0.495603	−	0.868549i	$-0.334947\pi$
$409$	8198.77	0.991205	0.495603	+	0.868549i	$0.334947\pi$
$410$	−3057.02	−0.368233
$411$	0	0
$412$	7338.11	0.877482
$413$	−3665.39	−0.436712
$414$	0	0
$415$	2800.25	0.331226
$416$	5352.59	0.630847
$417$	0	0
$418$	5964.41	0.697916
$419$	3509.69	0.409212	0.204606	−	0.978844i	$-0.434409\pi$
$419$	3509.69	0.409212	0.204606	+	0.978844i	$0.434409\pi$
$420$	0	0
$421$	−1228.24	−0.142188	−0.0710938	−	0.997470i	$-0.522649\pi$
$421$	−1228.24	−0.142188	−0.0710938	+	0.997470i	$0.522649\pi$
$422$	2179.09	0.251365
$423$	0	0
$424$	−17239.1	−1.97454
$425$	7509.29	0.857068
$426$	0	0
$427$	−8700.23	−0.986027
$428$	−5625.86	−0.635365
$429$	0	0
$430$	−2133.66	−0.239289
$431$	15746.8	1.75985	0.879926	−	0.475111i	$-0.157592\pi$
$431$	15746.8	1.75985	0.879926	+	0.475111i	$0.157592\pi$
$432$	0	0
$433$	15416.5	1.71101	0.855507	−	0.517792i	$-0.173246\pi$
$433$	15416.5	1.71101	0.855507	+	0.517792i	$0.173246\pi$
$434$	540.539	0.0597851
$435$	0	0
$436$	6635.51	0.728860
$437$	6083.57	0.665941
$438$	0	0
$439$	−13029.2	−1.41651	−0.708256	−	0.705956i	$-0.750518\pi$
$439$	−13029.2	−1.41651	−0.708256	+	0.705956i	$0.750518\pi$
$440$	3014.84	0.326652
$441$	0	0
$442$	−4427.58	−0.476467
$443$	−8891.43	−0.953599	−0.476800	−	0.879012i	$-0.658203\pi$
$443$	−8891.43	−0.953599	−0.476800	+	0.879012i	$0.658203\pi$
$444$	0	0
$445$	−3180.92	−0.338854
$446$	−9167.90	−0.973346
$447$	0	0
$448$	−5835.61	−0.615417
$449$	12338.8	1.29689	0.648447	−	0.761260i	$-0.275419\pi$
$449$	12338.8	1.29689	0.648447	+	0.761260i	$0.275419\pi$
$450$	0	0
$451$	9789.60	1.02212
$452$	5593.76	0.582098
$453$	0	0
$454$	−9482.07	−0.980211
$455$	−2047.88	−0.211003
$456$	0	0
$457$	−14357.2	−1.46958	−0.734791	−	0.678293i	$-0.762720\pi$
$457$	−14357.2	−1.46958	−0.734791	+	0.678293i	$0.762720\pi$
$458$	10433.3	1.06445
$459$	0	0
$460$	1073.02	0.108761
$461$	−13873.5	−1.40164	−0.700818	−	0.713341i	$-0.747181\pi$
$461$	−13873.5	−1.40164	−0.700818	+	0.713341i	$0.747181\pi$
$462$	0	0
$463$	6720.52	0.674577	0.337288	−	0.941401i	$-0.390490\pi$
$463$	6720.52	0.674577	0.337288	+	0.941401i	$0.390490\pi$
$464$	−137.246	−0.0137316
$465$	0	0
$466$	−9304.36	−0.924927
$467$	−13853.0	−1.37267	−0.686337	−	0.727284i	$-0.740782\pi$
$467$	−13853.0	−1.37267	−0.686337	+	0.727284i	$0.740782\pi$
$468$	0	0
$469$	487.740	0.0480208
$470$	−845.222	−0.0829515
$471$	0	0
$472$	6148.01	0.599545
$473$	6832.70	0.664203
$474$	0	0
$475$	−11525.4	−1.11331
$476$	−4354.73	−0.419325
$477$	0	0
$478$	460.489	0.0440634
$479$	−6813.41	−0.649922	−0.324961	−	0.945727i	$-0.605351\pi$
$479$	−6813.41	−0.649922	−0.324961	+	0.945727i	$0.605351\pi$
$480$	0	0
$481$	7785.41	0.738012
$482$	−9727.76	−0.919269
$483$	0	0
$484$	2338.06	0.219577
$485$	7880.56	0.737810
$486$	0	0
$487$	6464.04	0.601465	0.300733	−	0.953708i	$-0.402769\pi$
$487$	6464.04	0.601465	0.300733	+	0.953708i	$0.402769\pi$
$488$	14593.0	1.35368
$489$	0	0
$490$	−1258.43	−0.116021
$491$	−4063.25	−0.373467	−0.186733	−	0.982411i	$-0.559790\pi$
$491$	−4063.25	−0.373467	−0.186733	+	0.982411i	$0.559790\pi$
$492$	0	0
$493$	−872.852	−0.0797389
$494$	6795.53	0.618917
$495$	0	0
$496$	−224.878	−0.0203575
$497$	−1554.32	−0.140284
$498$	0	0
$499$	15417.9	1.38317	0.691583	−	0.722297i	$-0.256913\pi$
$499$	15417.9	1.38317	0.691583	+	0.722297i	$0.256913\pi$
$500$	−4467.72	−0.399605
$501$	0	0
$502$	−13442.8	−1.19518
$503$	−10985.5	−0.973792	−0.486896	−	0.873460i	$-0.661871\pi$
$503$	−10985.5	−0.973792	−0.486896	+	0.873460i	$0.661871\pi$
$504$	0	0
$505$	−563.597	−0.0496629
$506$	2975.05	0.261377
$507$	0	0
$508$	−2056.35	−0.179598
$509$	12010.2	1.04586	0.522929	−	0.852376i	$-0.324839\pi$
$509$	12010.2	1.04586	0.522929	+	0.852376i	$0.324839\pi$
$510$	0	0
$511$	12828.3	1.11055
$512$	4039.15	0.348646
$513$	0	0
$514$	14642.6	1.25653
$515$	7775.02	0.665259
$516$	0	0
$517$	2706.68	0.230251
$518$	−6629.73	−0.562343
$519$	0	0
$520$	3434.94	0.289677
$521$	−6895.43	−0.579836	−0.289918	−	0.957052i	$-0.593628\pi$
$521$	−6895.43	−0.579836	−0.289918	+	0.957052i	$0.593628\pi$
$522$	0	0
$523$	16667.0	1.39349	0.696747	−	0.717317i	$-0.254630\pi$
$523$	16667.0	1.39349	0.696747	+	0.717317i	$0.254630\pi$
$524$	−7816.79	−0.651676
$525$	0	0
$526$	−3065.05	−0.254073
$527$	−1430.17	−0.118215
$528$	0	0
$529$	−9132.52	−0.750598
$530$	−6373.62	−0.522363
$531$	0	0
$532$	6683.72	0.544692
$533$	11153.7	0.906420
$534$	0	0
$535$	−5960.83	−0.481699
$536$	−818.093	−0.0659258
$537$	0	0
$538$	8200.07	0.657119
$539$	4029.92	0.322043
$540$	0	0
$541$	5011.21	0.398242	0.199121	−	0.979975i	$-0.436191\pi$
$541$	5011.21	0.398242	0.199121	+	0.979975i	$0.436191\pi$
$542$	−16013.8	−1.26910
$543$	0	0
$544$	12059.7	0.950473
$545$	7030.59	0.552582
$546$	0	0
$547$	22306.8	1.74364	0.871819	−	0.489828i	$-0.162940\pi$
$547$	22306.8	1.74364	0.871819	+	0.489828i	$0.162940\pi$
$548$	8091.36	0.630740
$549$	0	0
$550$	−5636.26	−0.436965
$551$	1339.67	0.103579
$552$	0	0
$553$	−1165.73	−0.0896416
$554$	−7491.79	−0.574541
$555$	0	0
$556$	−1995.03	−0.152173
$557$	−8349.63	−0.635162	−0.317581	−	0.948231i	$-0.602871\pi$
$557$	−8349.63	−0.635162	−0.317581	+	0.948231i	$0.602871\pi$
$558$	0	0
$559$	7784.81	0.589020
$560$	−725.501	−0.0547464
$561$	0	0
$562$	−4530.75	−0.340068
$563$	−10064.6	−0.753417	−0.376708	−	0.926332i	$-0.622944\pi$
$563$	−10064.6	−0.753417	−0.376708	+	0.926332i	$0.622944\pi$
$564$	0	0
$565$	5926.81	0.441315
$566$	3526.02	0.261854
$567$	0	0
$568$	2607.09	0.192590
$569$	1651.40	0.121670	0.0608350	−	0.998148i	$-0.480624\pi$
$569$	1651.40	0.121670	0.0608350	+	0.998148i	$0.480624\pi$
$570$	0	0
$571$	23898.3	1.75151	0.875757	−	0.482752i	$-0.160363\pi$
$571$	23898.3	1.75151	0.875757	+	0.482752i	$0.160363\pi$
$572$	−3838.31	−0.280573
$573$	0	0
$574$	−9498.05	−0.690664
$575$	−5748.86	−0.416946
$576$	0	0
$577$	10181.4	0.734586	0.367293	−	0.930105i	$-0.380285\pi$
$577$	10181.4	0.734586	0.367293	+	0.930105i	$0.380285\pi$
$578$	−509.598	−0.0366721
$579$	0	0
$580$	236.291	0.0169163
$581$	8700.27	0.621253
$582$	0	0
$583$	20410.4	1.44994
$584$	−21517.1	−1.52463
$585$	0	0
$586$	−9516.44	−0.670854
$587$	9971.33	0.701126	0.350563	−	0.936539i	$-0.385990\pi$
$587$	9971.33	0.701126	0.350563	+	0.936539i	$0.385990\pi$
$588$	0	0
$589$	2195.05	0.153557
$590$	2273.03	0.158609
$591$	0	0
$592$	2758.13	0.191484
$593$	−16799.7	−1.16337	−0.581686	−	0.813414i	$-0.697607\pi$
$593$	−16799.7	−1.16337	−0.581686	+	0.813414i	$0.697607\pi$
$594$	0	0
$595$	−4614.02	−0.317910
$596$	−10667.3	−0.733140
$597$	0	0
$598$	3389.60	0.231791
$599$	4059.29	0.276892	0.138446	−	0.990370i	$-0.455789\pi$
$599$	4059.29	0.276892	0.138446	+	0.990370i	$0.455789\pi$
$600$	0	0
$601$	−2424.31	−0.164542	−0.0822711	−	0.996610i	$-0.526217\pi$
$601$	−2424.31	−0.164542	−0.0822711	+	0.996610i	$0.526217\pi$
$602$	−6629.21	−0.448815
$603$	0	0
$604$	14251.3	0.960062
$605$	2477.27	0.166471
$606$	0	0
$607$	17901.3	1.19702	0.598510	−	0.801115i	$-0.295760\pi$
$607$	17901.3	1.19702	0.598510	+	0.801115i	$0.295760\pi$
$608$	−18509.5	−1.23464
$609$	0	0
$610$	5395.31	0.358114
$611$	3083.85	0.204188
$612$	0	0
$613$	3978.84	0.262160	0.131080	−	0.991372i	$-0.458156\pi$
$613$	3978.84	0.262160	0.131080	+	0.991372i	$0.458156\pi$
$614$	9792.03	0.643606
$615$	0	0
$616$	9366.99	0.612673
$617$	14969.5	0.976741	0.488371	−	0.872636i	$-0.337591\pi$
$617$	14969.5	0.976741	0.488371	+	0.872636i	$0.337591\pi$
$618$	0	0
$619$	−19925.6	−1.29383	−0.646913	−	0.762564i	$-0.723940\pi$
$619$	−19925.6	−1.29383	−0.646913	+	0.762564i	$0.723940\pi$
$620$	387.163	0.0250788
$621$	0	0
$622$	481.258	0.0310236
$623$	−9882.99	−0.635560
$624$	0	0
$625$	8311.43	0.531931
$626$	−2573.58	−0.164315
$627$	0	0
$628$	3088.93	0.196277
$629$	17541.0	1.11194
$630$	0	0
$631$	−14340.9	−0.904757	−0.452379	−	0.891826i	$-0.649424\pi$
$631$	−14340.9	−0.904757	−0.452379	+	0.891826i	$0.649424\pi$
$632$	1955.29	0.123065
$633$	0	0
$634$	−7898.65	−0.494788
$635$	−2178.78	−0.136161
$636$	0	0
$637$	4591.47	0.285590
$638$	655.138	0.0406539
$639$	0	0
$640$	−2472.44	−0.152706
$641$	−3101.58	−0.191116	−0.0955578	−	0.995424i	$-0.530463\pi$
$641$	−3101.58	−0.191116	−0.0955578	+	0.995424i	$0.530463\pi$
$642$	0	0
$643$	−24212.2	−1.48497	−0.742486	−	0.669861i	$-0.766354\pi$
$643$	−24212.2	−1.48497	−0.742486	+	0.669861i	$0.766354\pi$
$644$	3333.84	0.203993
$645$	0	0
$646$	15310.8	0.932499
$647$	14136.6	0.858992	0.429496	−	0.903069i	$-0.358691\pi$
$647$	14136.6	0.858992	0.429496	+	0.903069i	$0.358691\pi$
$648$	0	0
$649$	−7279.01	−0.440256
$650$	−6421.64	−0.387504
$651$	0	0
$652$	−4751.84	−0.285424
$653$	−8407.94	−0.503872	−0.251936	−	0.967744i	$-0.581067\pi$
$653$	−8407.94	−0.503872	−0.251936	+	0.967744i	$0.581067\pi$
$654$	0	0
$655$	−8282.21	−0.494065
$656$	3951.42	0.235178
$657$	0	0
$658$	−2626.08	−0.155585
$659$	406.791	0.0240460	0.0120230	−	0.999928i	$-0.496173\pi$
$659$	406.791	0.0240460	0.0120230	+	0.999928i	$0.496173\pi$
$660$	0	0
$661$	−22192.4	−1.30588	−0.652939	−	0.757410i	$-0.726464\pi$
$661$	−22192.4	−1.30588	−0.652939	+	0.757410i	$0.726464\pi$
$662$	1357.66	0.0797086
$663$	0	0
$664$	−14593.1	−0.852893
$665$	7081.67	0.412955
$666$	0	0
$667$	668.226	0.0387913
$668$	7189.55	0.416425
$669$	0	0
$670$	−302.464	−0.0174406
$671$	−17277.6	−0.994029
$672$	0	0
$673$	−17824.1	−1.02090	−0.510452	−	0.859906i	$-0.670522\pi$
$673$	−17824.1	−1.02090	−0.510452	+	0.859906i	$0.670522\pi$
$674$	7586.60	0.433568
$675$	0	0
$676$	5046.92	0.287148
$677$	−27332.0	−1.55163	−0.775815	−	0.630961i	$-0.782661\pi$
$677$	−27332.0	−1.55163	−0.775815	+	0.630961i	$0.782661\pi$
$678$	0	0
$679$	24484.6	1.38385
$680$	7739.15	0.436446
$681$	0	0
$682$	1073.44	0.0602702
$683$	−31853.8	−1.78456	−0.892278	−	0.451486i	$-0.850894\pi$
$683$	−31853.8	−1.78456	−0.892278	+	0.451486i	$0.850894\pi$
$684$	0	0
$685$	8573.12	0.478193
$686$	−13238.0	−0.736778
$687$	0	0
$688$	2757.91	0.152826
$689$	23254.5	1.28582
$690$	0	0
$691$	24069.9	1.32513	0.662564	−	0.749006i	$-0.269468\pi$
$691$	24069.9	1.32513	0.662564	+	0.749006i	$0.269468\pi$
$692$	16723.1	0.918667
$693$	0	0
$694$	−8070.40	−0.441424
$695$	−2113.81	−0.115369
$696$	0	0
$697$	25130.1	1.36567
$698$	10833.1	0.587450
$699$	0	0
$700$	−6315.99	−0.341031
$701$	610.991	0.0329199	0.0164599	−	0.999865i	$-0.494760\pi$
$701$	610.991	0.0329199	0.0164599	+	0.999865i	$0.494760\pi$
$702$	0	0
$703$	−26922.3	−1.44437
$704$	−11588.8	−0.620411
$705$	0	0
$706$	20752.3	1.10627
$707$	−1751.08	−0.0931485
$708$	0	0
$709$	−15273.0	−0.809014	−0.404507	−	0.914535i	$-0.632557\pi$
$709$	−15273.0	−0.809014	−0.404507	+	0.914535i	$0.632557\pi$
$710$	963.888	0.0509494
$711$	0	0
$712$	16576.9	0.872535
$713$	1094.89	0.0575090
$714$	0	0
$715$	−4066.84	−0.212715
$716$	−2846.08	−0.148552
$717$	0	0
$718$	−8966.49	−0.466054
$719$	−27319.3	−1.41702	−0.708510	−	0.705700i	$-0.750632\pi$
$719$	−27319.3	−1.41702	−0.708510	+	0.705700i	$0.750632\pi$
$720$	0	0
$721$	24156.7	1.24777
$722$	−10283.8	−0.530087
$723$	0	0
$724$	−8998.67	−0.461924
$725$	−1265.96	−0.0648505
$726$	0	0
$727$	15530.0	0.792262	0.396131	−	0.918194i	$-0.370353\pi$
$727$	15530.0	0.792262	0.396131	+	0.918194i	$0.370353\pi$
$728$	10672.2	0.543323
$729$	0	0
$730$	−7955.28	−0.403340
$731$	17539.7	0.887454
$732$	0	0
$733$	18585.2	0.936507	0.468254	−	0.883594i	$-0.344883\pi$
$733$	18585.2	0.936507	0.468254	+	0.883594i	$0.344883\pi$
$734$	23508.4	1.18217
$735$	0	0
$736$	−9232.53	−0.462385
$737$	968.591	0.0484105
$738$	0	0
$739$	−27867.1	−1.38715	−0.693577	−	0.720382i	$-0.743966\pi$
$739$	−27867.1	−1.38715	−0.693577	+	0.720382i	$0.743966\pi$
$740$	−4748.56	−0.235893
$741$	0	0
$742$	−19802.6	−0.979752
$743$	−26465.7	−1.30677	−0.653387	−	0.757024i	$-0.726653\pi$
$743$	−26465.7	−1.30677	−0.653387	+	0.757024i	$0.726653\pi$
$744$	0	0
$745$	−11302.5	−0.555826
$746$	−9720.52	−0.477069
$747$	0	0
$748$	−8647.96	−0.422728
$749$	−18520.1	−0.903483
$750$	0	0
$751$	−17010.6	−0.826533	−0.413267	−	0.910610i	$-0.635612\pi$
$751$	−17010.6	−0.826533	−0.413267	+	0.910610i	$0.635612\pi$
$752$	1092.51	0.0529784
$753$	0	0
$754$	746.428	0.0360522
$755$	15099.8	0.727866
$756$	0	0
$757$	−16202.7	−0.777934	−0.388967	−	0.921252i	$-0.627168\pi$
$757$	−16202.7	−0.777934	−0.388967	+	0.921252i	$0.627168\pi$
$758$	9633.12	0.461597
$759$	0	0
$760$	−11878.2	−0.566930
$761$	13718.1	0.653457	0.326729	−	0.945118i	$-0.394054\pi$
$761$	13718.1	0.653457	0.326729	+	0.945118i	$0.394054\pi$
$762$	0	0
$763$	21843.8	1.03643
$764$	−1296.61	−0.0613999
$765$	0	0
$766$	−20320.2	−0.958482
$767$	−8293.31	−0.390422
$768$	0	0
$769$	−19139.3	−0.897505	−0.448752	−	0.893656i	$-0.648131\pi$
$769$	−19139.3	−0.897505	−0.448752	+	0.893656i	$0.648131\pi$
$770$	3463.15	0.162082
$771$	0	0
$772$	16112.6	0.751173
$773$	1454.06	0.0676572	0.0338286	−	0.999428i	$-0.489230\pi$
$773$	1454.06	0.0676572	0.0338286	+	0.999428i	$0.489230\pi$
$774$	0	0
$775$	−2074.28	−0.0961423
$776$	−41068.4	−1.89983
$777$	0	0
$778$	−22974.1	−1.05869
$779$	−38570.1	−1.77396
$780$	0	0
$781$	−3086.69	−0.141422
$782$	7637.01	0.349231
$783$	0	0
$784$	1626.62	0.0740988
$785$	3272.85	0.148806
$786$	0	0
$787$	−34703.9	−1.57187	−0.785934	−	0.618311i	$-0.787817\pi$
$787$	−34703.9	−1.57187	−0.785934	+	0.618311i	$0.787817\pi$
$788$	−12831.6	−0.580086
$789$	0	0
$790$	722.907	0.0325568
$791$	18414.4	0.827738
$792$	0	0
$793$	−19685.1	−0.881512
$794$	−7836.47	−0.350260
$795$	0	0
$796$	2462.75	0.109660
$797$	−5327.34	−0.236768	−0.118384	−	0.992968i	$-0.537771\pi$
$797$	−5327.34	−0.236768	−0.118384	+	0.992968i	$0.537771\pi$
$798$	0	0
$799$	6948.12	0.307643
$800$	17491.1	0.773006
$801$	0	0
$802$	−27488.6	−1.21029
$803$	25475.5	1.11956
$804$	0	0
$805$	3532.33	0.154656
$806$	1223.02	0.0534481
$807$	0	0
$808$	2937.10	0.127880
$809$	23677.3	1.02899	0.514494	−	0.857494i	$-0.327980\pi$
$809$	23677.3	1.02899	0.514494	+	0.857494i	$0.327980\pi$
$810$	0	0
$811$	−24380.5	−1.05563	−0.527814	−	0.849360i	$-0.676988\pi$
$811$	−24380.5	−1.05563	−0.527814	+	0.849360i	$0.676988\pi$
$812$	734.148	0.0317285
$813$	0	0
$814$	−13165.8	−0.566906
$815$	−5034.76	−0.216393
$816$	0	0
$817$	−26920.2	−1.15278
$818$	−15796.8	−0.675212
$819$	0	0
$820$	−6803.01	−0.289721
$821$	21296.0	0.905283	0.452641	−	0.891693i	$-0.350482\pi$
$821$	21296.0	0.905283	0.452641	+	0.891693i	$0.350482\pi$
$822$	0	0
$823$	4908.88	0.207913	0.103957	−	0.994582i	$-0.466850\pi$
$823$	4908.88	0.207913	0.103957	+	0.994582i	$0.466850\pi$
$824$	−40518.4	−1.71301
$825$	0	0
$826$	7062.23	0.297490
$827$	38572.6	1.62189	0.810944	−	0.585124i	$-0.198954\pi$
$827$	38572.6	1.62189	0.810944	+	0.585124i	$0.198954\pi$
$828$	0	0
$829$	30118.0	1.26181	0.630905	−	0.775860i	$-0.282684\pi$
$829$	30118.0	1.26181	0.630905	+	0.775860i	$0.282684\pi$
$830$	−5395.33	−0.225632
$831$	0	0
$832$	−13203.7	−0.550185
$833$	10344.9	0.430288
$834$	0	0
$835$	7617.62	0.315711
$836$	13273.0	0.549112
$837$	0	0
$838$	−6762.25	−0.278756
$839$	−24615.7	−1.01291	−0.506454	−	0.862267i	$-0.669044\pi$
$839$	−24615.7	−1.01291	−0.506454	+	0.862267i	$0.669044\pi$
$840$	0	0
$841$	−24241.8	−0.993967
$842$	2366.50	0.0968586
$843$	0	0
$844$	4849.28	0.197771
$845$	5347.41	0.217700
$846$	0	0
$847$	7696.78	0.312237
$848$	8238.36	0.333616
$849$	0	0
$850$	−14468.4	−0.583838
$851$	−13428.8	−0.540933
$852$	0	0
$853$	24728.8	0.992614	0.496307	−	0.868147i	$-0.334689\pi$
$853$	24728.8	0.992614	0.496307	+	0.868147i	$0.334689\pi$
$854$	16763.0	0.671685
$855$	0	0
$856$	31064.0	1.24036
$857$	1605.12	0.0639788	0.0319894	−	0.999488i	$-0.489816\pi$
$857$	1605.12	0.0639788	0.0319894	+	0.999488i	$0.489816\pi$
$858$	0	0
$859$	−43424.7	−1.72483	−0.862417	−	0.506199i	$-0.831050\pi$
$859$	−43424.7	−1.72483	−0.862417	+	0.506199i	$0.831050\pi$
$860$	−4748.19	−0.188270
$861$	0	0
$862$	−30339.9	−1.19882
$863$	28312.7	1.11678	0.558388	−	0.829580i	$-0.311420\pi$
$863$	28312.7	1.11678	0.558388	+	0.829580i	$0.311420\pi$
$864$	0	0
$865$	17718.8	0.696483
$866$	−29703.4	−1.16555
$867$	0	0
$868$	1202.90	0.0470382
$869$	−2314.99	−0.0903690
$870$	0	0
$871$	1103.56	0.0429308
$872$	−36638.9	−1.42288
$873$	0	0
$874$	−11721.4	−0.453641
$875$	−14707.5	−0.568234
$876$	0	0
$877$	16723.1	0.643897	0.321948	−	0.946757i	$-0.395662\pi$
$877$	16723.1	0.643897	0.321948	+	0.946757i	$0.395662\pi$
$878$	25103.7	0.964932
$879$	0	0
$880$	−1440.75	−0.0551907
$881$	−986.914	−0.0377412	−0.0188706	−	0.999822i	$-0.506007\pi$
$881$	−986.914	−0.0377412	−0.0188706	+	0.999822i	$0.506007\pi$
$882$	0	0
$883$	21840.2	0.832368	0.416184	−	0.909280i	$-0.363367\pi$
$883$	21840.2	0.832368	0.416184	+	0.909280i	$0.363367\pi$
$884$	−9853.01	−0.374879
$885$	0	0
$886$	17131.4	0.649595
$887$	−629.867	−0.0238431	−0.0119216	−	0.999929i	$-0.503795\pi$
$887$	−629.867	−0.0238431	−0.0119216	+	0.999929i	$0.503795\pi$
$888$	0	0
$889$	−6769.40	−0.255386
$890$	6128.78	0.230828
$891$	0	0
$892$	−20402.0	−0.765817
$893$	−10664.1	−0.399619
$894$	0	0
$895$	−3015.54	−0.112624
$896$	−7681.79	−0.286418
$897$	0	0
$898$	−23773.6	−0.883449
$899$	241.107	0.00894478
$900$	0	0
$901$	52394.1	1.93729
$902$	−18862.0	−0.696269
$903$	0	0
$904$	−30886.7	−1.13637
$905$	−9534.45	−0.350205
$906$	0	0
$907$	34384.5	1.25879	0.629393	−	0.777088i	$-0.283304\pi$
$907$	34384.5	1.25879	0.629393	+	0.777088i	$0.283304\pi$
$908$	−21101.1	−0.771218
$909$	0	0
$910$	3945.72	0.143736
$911$	−33862.9	−1.23153	−0.615767	−	0.787928i	$-0.711154\pi$
$911$	−33862.9	−1.23153	−0.615767	+	0.787928i	$0.711154\pi$
$912$	0	0
$913$	17277.6	0.626294
$914$	27662.4	1.00108
$915$	0	0
$916$	23218.1	0.837496
$917$	−25732.5	−0.926677
$918$	0	0
$919$	−26656.8	−0.956830	−0.478415	−	0.878134i	$-0.658788\pi$
$919$	−26656.8	−0.956830	−0.478415	+	0.878134i	$0.658788\pi$
$920$	−5924.83	−0.212322
$921$	0	0
$922$	26730.6	0.954798
$923$	−3516.81	−0.125414
$924$	0	0
$925$	25441.1	0.904321
$926$	−12948.7	−0.459524
$927$	0	0
$928$	−2033.11	−0.0719181
$929$	12246.8	0.432514	0.216257	−	0.976336i	$-0.430615\pi$
$929$	12246.8	0.432514	0.216257	+	0.976336i	$0.430615\pi$
$930$	0	0
$931$	−15877.5	−0.558931
$932$	−20705.7	−0.727721
$933$	0	0
$934$	26691.0	0.935070
$935$	−9162.86	−0.320490
$936$	0	0
$937$	−11620.7	−0.405155	−0.202577	−	0.979266i	$-0.564932\pi$
$937$	−11620.7	−0.405155	−0.202577	+	0.979266i	$0.564932\pi$
$938$	−939.745	−0.0327119
$939$	0	0
$940$	−1880.93	−0.0652652
$941$	50209.2	1.73940	0.869698	−	0.493584i	$-0.164313\pi$
$941$	50209.2	1.73940	0.869698	+	0.493584i	$0.164313\pi$
$942$	0	0
$943$	−19238.8	−0.664369
$944$	−2938.06	−0.101298
$945$	0	0
$946$	−13164.8	−0.452457
$947$	28445.8	0.976097	0.488049	−	0.872816i	$-0.337709\pi$
$947$	28445.8	0.976097	0.488049	+	0.872816i	$0.337709\pi$
$948$	0	0
$949$	29025.4	0.992838
$950$	22206.3	0.758388
$951$	0	0
$952$	24045.3	0.818605
$953$	38782.3	1.31824	0.659119	−	0.752038i	$-0.270929\pi$
$953$	38782.3	1.31824	0.659119	+	0.752038i	$0.270929\pi$
$954$	0	0
$955$	−1373.81	−0.0465501
$956$	1024.76	0.0346685
$957$	0	0
$958$	13127.6	0.442729
$959$	26636.4	0.896907
$960$	0	0
$961$	−29395.9	−0.986739
$962$	−15000.4	−0.502736
$963$	0	0
$964$	−21647.9	−0.723269
$965$	17072.0	0.569498
$966$	0	0
$967$	54398.6	1.80904	0.904520	−	0.426432i	$-0.140230\pi$
$967$	54398.6	1.80904	0.904520	+	0.426432i	$0.140230\pi$
$968$	−12909.9	−0.428657
$969$	0	0
$970$	−15183.7	−0.502598
$971$	2476.35	0.0818432	0.0409216	−	0.999162i	$-0.486971\pi$
$971$	2476.35	0.0818432	0.0409216	+	0.999162i	$0.486971\pi$
$972$	0	0
$973$	−6567.54	−0.216388
$974$	−12454.5	−0.409720
$975$	0	0
$976$	−6973.83	−0.228716
$977$	−48041.8	−1.57318	−0.786588	−	0.617478i	$-0.788154\pi$
$977$	−48041.8	−1.57318	−0.786588	+	0.617478i	$0.788154\pi$
$978$	0	0
$979$	−19626.4	−0.640717
$980$	−2800.48	−0.0912838
$981$	0	0
$982$	7828.81	0.254407
$983$	−3188.75	−0.103464	−0.0517322	−	0.998661i	$-0.516474\pi$
$983$	−3188.75	−0.103464	−0.0517322	+	0.998661i	$0.516474\pi$
$984$	0	0
$985$	−13595.6	−0.439789
$986$	1681.75	0.0543184
$987$	0	0
$988$	15122.6	0.486957
$989$	−13427.8	−0.431728
$990$	0	0
$991$	−48105.8	−1.54201	−0.771005	−	0.636829i	$-0.780246\pi$
$991$	−48105.8	−1.54201	−0.771005	+	0.636829i	$0.780246\pi$
$992$	−3331.24	−0.106620
$993$	0	0
$994$	2994.77	0.0955616
$995$	2609.38	0.0831385
$996$	0	0
$997$	54934.0	1.74501	0.872507	−	0.488602i	$-0.162493\pi$
$997$	54934.0	1.74501	0.872507	+	0.488602i	$0.162493\pi$
$998$	−29706.2	−0.942218
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.h.1.20	yes	59
3.2	odd	2		2151.4.a.g.1.40	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.40	✓	59	3.2	odd	2
2151.4.a.h.1.20	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)

\(f(q)\)	\(=\)	\(q-1.92673 q^{2} -4.28770 q^{4} -4.54299 q^{5} -14.1149 q^{7} +23.6751 q^{8} +O(q^{10})\)	\(q-1.92673 q^{2} -4.28770 q^{4} -4.54299 q^{5} -14.1149 q^{7} +23.6751 q^{8} +8.75313 q^{10} -28.0304 q^{11} -31.9364 q^{13} +27.1957 q^{14} -11.3141 q^{16} -71.9548 q^{17} +110.437 q^{19} +19.4790 q^{20} +54.0072 q^{22} +55.0861 q^{23} -104.361 q^{25} +61.5328 q^{26} +60.5204 q^{28} +12.1306 q^{29} +19.8759 q^{31} -167.602 q^{32} +138.638 q^{34} +64.1239 q^{35} -243.779 q^{37} -212.783 q^{38} -107.556 q^{40} -349.249 q^{41} -243.760 q^{43} +120.186 q^{44} -106.136 q^{46} -96.5623 q^{47} -143.769 q^{49} +201.076 q^{50} +136.933 q^{52} -728.153 q^{53} +127.342 q^{55} -334.172 q^{56} -23.3724 q^{58} +259.682 q^{59} +616.386 q^{61} -38.2956 q^{62} +413.436 q^{64} +145.087 q^{65} -34.5550 q^{67} +308.520 q^{68} -123.550 q^{70} +110.119 q^{71} -908.850 q^{73} +469.697 q^{74} -473.522 q^{76} +395.647 q^{77} +82.5884 q^{79} +51.3996 q^{80} +672.910 q^{82} -616.389 q^{83} +326.890 q^{85} +469.661 q^{86} -663.624 q^{88} +700.181 q^{89} +450.779 q^{91} -236.193 q^{92} +186.050 q^{94} -501.716 q^{95} -1734.66 q^{97} +277.005 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) \) 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

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