LMFDB - Embedded newform 2151.4.a.a.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$126.913108422$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q+1.51884 q^{2} -5.69312 q^{4} +1.30911 q^{5} -3.41938 q^{7} -20.7977 q^{8} +O(q^{10})$	\(q+1.51884 q^{2} -5.69312 q^{4} +1.30911 q^{5} -3.41938 q^{7} -20.7977 q^{8} +1.98833 q^{10} -12.1993 q^{11} +42.4318 q^{13} -5.19350 q^{14} +13.9567 q^{16} +58.4877 q^{17} -36.2996 q^{19} -7.45292 q^{20} -18.5288 q^{22} -142.786 q^{23} -123.286 q^{25} +64.4470 q^{26} +19.4670 q^{28} +94.7940 q^{29} +304.744 q^{31} +187.579 q^{32} +88.8334 q^{34} -4.47635 q^{35} +29.1316 q^{37} -55.1333 q^{38} -27.2264 q^{40} +319.989 q^{41} +106.095 q^{43} +69.4520 q^{44} -216.869 q^{46} -71.3313 q^{47} -331.308 q^{49} -187.252 q^{50} -241.569 q^{52} -184.653 q^{53} -15.9702 q^{55} +71.1152 q^{56} +143.977 q^{58} -620.066 q^{59} -641.669 q^{61} +462.857 q^{62} +173.250 q^{64} +55.5478 q^{65} +157.098 q^{67} -332.978 q^{68} -6.79886 q^{70} +868.090 q^{71} +325.145 q^{73} +44.2463 q^{74} +206.658 q^{76} +41.7140 q^{77} +376.024 q^{79} +18.2708 q^{80} +486.013 q^{82} +411.096 q^{83} +76.5668 q^{85} +161.141 q^{86} +253.717 q^{88} -409.934 q^{89} -145.090 q^{91} +812.899 q^{92} -108.341 q^{94} -47.5201 q^{95} -376.544 q^{97} -503.204 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.51884	0.536991	0.268496	−	0.963281i	$-0.413474\pi$
$2$	1.51884	0.536991	0.268496	+	0.963281i	$0.413474\pi$
$3$	0	0
$3$	0	0
$4$	−5.69312	−0.711641
$5$	1.30911	0.117090	0.0585452	−	0.998285i	$-0.481354\pi$
$5$	1.30911	0.117090	0.0585452	+	0.998285i	$0.481354\pi$
$6$	0	0
$7$	−3.41938	−0.184629	−0.0923147	−	0.995730i	$-0.529427\pi$
$7$	−3.41938	−0.184629	−0.0923147	+	0.995730i	$0.529427\pi$
$8$	−20.7977	−0.919136
$9$	0	0
$10$	1.98833	0.0628765
$11$	−12.1993	−0.334384	−0.167192	−	0.985924i	$-0.553470\pi$
$11$	−12.1993	−0.334384	−0.167192	+	0.985924i	$0.553470\pi$
$12$	0	0
$13$	42.4318	0.905265	0.452633	−	0.891697i	$-0.350485\pi$
$13$	42.4318	0.905265	0.452633	+	0.891697i	$0.350485\pi$
$14$	−5.19350	−0.0991443
$15$	0	0
$16$	13.9567	0.218073
$17$	58.4877	0.834432	0.417216	−	0.908807i	$-0.363006\pi$
$17$	58.4877	0.834432	0.417216	+	0.908807i	$0.363006\pi$
$18$	0	0
$19$	−36.2996	−0.438300	−0.219150	−	0.975691i	$-0.570328\pi$
$19$	−36.2996	−0.438300	−0.219150	+	0.975691i	$0.570328\pi$
$20$	−7.45292	−0.0833262
$21$	0	0
$22$	−18.5288	−0.179561
$23$	−142.786	−1.29448	−0.647238	−	0.762288i	$-0.724076\pi$
$23$	−142.786	−1.29448	−0.647238	+	0.762288i	$0.724076\pi$
$24$	0	0
$25$	−123.286	−0.986290
$26$	64.4470	0.486119
$27$	0	0
$28$	19.4670	0.131390
$29$	94.7940	0.606993	0.303497	−	0.952833i	$-0.401846\pi$
$29$	94.7940	0.606993	0.303497	+	0.952833i	$0.401846\pi$
$30$	0	0
$31$	304.744	1.76560	0.882800	−	0.469749i	$-0.155656\pi$
$31$	304.744	1.76560	0.882800	+	0.469749i	$0.155656\pi$
$32$	187.579	1.03624
$33$	0	0
$34$	88.8334	0.448083
$35$	−4.47635	−0.0216183
$36$	0	0
$37$	29.1316	0.129438	0.0647190	−	0.997904i	$-0.479385\pi$
$37$	29.1316	0.129438	0.0647190	+	0.997904i	$0.479385\pi$
$38$	−55.1333	−0.235363
$39$	0	0
$40$	−27.2264	−0.107622
$41$	319.989	1.21888	0.609439	−	0.792833i	$-0.291395\pi$
$41$	319.989	1.21888	0.609439	+	0.792833i	$0.291395\pi$
$42$	0	0
$43$	106.095	0.376263	0.188132	−	0.982144i	$-0.439757\pi$
$43$	106.095	0.376263	0.188132	+	0.982144i	$0.439757\pi$
$44$	69.4520	0.237961
$45$	0	0
$46$	−216.869	−0.695122
$47$	−71.3313	−0.221378	−0.110689	−	0.993855i	$-0.535306\pi$
$47$	−71.3313	−0.221378	−0.110689	+	0.993855i	$0.535306\pi$
$48$	0	0
$49$	−331.308	−0.965912
$50$	−187.252	−0.529629
$51$	0	0
$52$	−241.569	−0.644224
$53$	−184.653	−0.478567	−0.239283	−	0.970950i	$-0.576912\pi$
$53$	−184.653	−0.478567	−0.239283	+	0.970950i	$0.576912\pi$
$54$	0	0
$55$	−15.9702	−0.0391531
$56$	71.1152	0.169699
$57$	0	0
$58$	143.977	0.325950
$59$	−620.066	−1.36823	−0.684116	−	0.729373i	$-0.739812\pi$
$59$	−620.066	−1.36823	−0.684116	+	0.729373i	$0.739812\pi$
$60$	0	0
$61$	−641.669	−1.34684	−0.673421	−	0.739260i	$-0.735176\pi$
$61$	−641.669	−1.34684	−0.673421	+	0.739260i	$0.735176\pi$
$62$	462.857	0.948111
$63$	0	0
$64$	173.250	0.338378
$65$	55.5478	0.105998
$66$	0	0
$67$	157.098	0.286456	0.143228	−	0.989690i	$-0.454252\pi$
$67$	157.098	0.286456	0.143228	+	0.989690i	$0.454252\pi$
$68$	−332.978	−0.593816
$69$	0	0
$70$	−6.79886	−0.0116088
$71$	868.090	1.45103	0.725517	−	0.688205i	$-0.241601\pi$
$71$	868.090	1.45103	0.725517	+	0.688205i	$0.241601\pi$
$72$	0	0
$73$	325.145	0.521306	0.260653	−	0.965433i	$-0.416062\pi$
$73$	325.145	0.521306	0.260653	+	0.965433i	$0.416062\pi$
$74$	44.2463	0.0695071
$75$	0	0
$76$	206.658	0.311912
$77$	41.7140	0.0617371
$78$	0	0
$79$	376.024	0.535519	0.267760	−	0.963486i	$-0.413717\pi$
$79$	376.024	0.535519	0.267760	+	0.963486i	$0.413717\pi$
$80$	18.2708	0.0255342
$81$	0	0
$82$	486.013	0.654526
$83$	411.096	0.543659	0.271830	−	0.962345i	$-0.412371\pi$
$83$	411.096	0.543659	0.271830	+	0.962345i	$0.412371\pi$
$84$	0	0
$85$	76.5668	0.0977039
$86$	161.141	0.202050
$87$	0	0
$88$	253.717	0.307344
$89$	−409.934	−0.488234	−0.244117	−	0.969746i	$-0.578498\pi$
$89$	−409.934	−0.488234	−0.244117	+	0.969746i	$0.578498\pi$
$90$	0	0
$91$	−145.090	−0.167139
$92$	812.899	0.921202
$93$	0	0
$94$	−108.341	−0.118878
$95$	−47.5201	−0.0513207
$96$	0	0
$97$	−376.544	−0.394147	−0.197074	−	0.980389i	$-0.563144\pi$
$97$	−376.544	−0.394147	−0.197074	+	0.980389i	$0.563144\pi$
$98$	−503.204	−0.518686
$99$	0	0
$100$	701.884	0.701884
$101$	1592.97	1.56937	0.784687	−	0.619893i	$-0.212824\pi$
$101$	1592.97	1.56937	0.784687	+	0.619893i	$0.212824\pi$
$102$	0	0
$103$	−1717.28	−1.64280	−0.821402	−	0.570349i	$-0.806808\pi$
$103$	−1717.28	−1.64280	−0.821402	+	0.570349i	$0.806808\pi$
$104$	−882.481	−0.832062
$105$	0	0
$106$	−280.458	−0.256986
$107$	−1050.79	−0.949383	−0.474692	−	0.880152i	$-0.657440\pi$
$107$	−1050.79	−0.949383	−0.474692	+	0.880152i	$0.657440\pi$
$108$	0	0
$109$	47.0484	0.0413433	0.0206717	−	0.999786i	$-0.493420\pi$
$109$	47.0484	0.0413433	0.0206717	+	0.999786i	$0.493420\pi$
$110$	−24.2562	−0.0210249
$111$	0	0
$112$	−47.7232	−0.0402627
$113$	323.209	0.269071	0.134535	−	0.990909i	$-0.457046\pi$
$113$	323.209	0.269071	0.134535	+	0.990909i	$0.457046\pi$
$114$	0	0
$115$	−186.923	−0.151571
$116$	−539.674	−0.431961
$117$	0	0
$118$	−941.780	−0.734728
$119$	−199.992	−0.154061
$120$	0	0
$121$	−1182.18	−0.888187
$122$	−974.593	−0.723242
$123$	0	0
$124$	−1734.94	−1.25647
$125$	−325.034	−0.232575
$126$	0	0
$127$	−1397.44	−0.976402	−0.488201	−	0.872731i	$-0.662347\pi$
$127$	−1397.44	−0.976402	−0.488201	+	0.872731i	$0.662347\pi$
$128$	−1237.50	−0.854533
$129$	0	0
$130$	84.3682	0.0569199
$131$	203.023	0.135406	0.0677031	−	0.997706i	$-0.478433\pi$
$131$	203.023	0.135406	0.0677031	+	0.997706i	$0.478433\pi$
$132$	0	0
$133$	124.122	0.0809230
$134$	238.606	0.153824
$135$	0	0
$136$	−1216.41	−0.766956
$137$	−1196.35	−0.746064	−0.373032	−	0.927818i	$-0.621682\pi$
$137$	−1196.35	−0.746064	−0.373032	+	0.927818i	$0.621682\pi$
$138$	0	0
$139$	−994.154	−0.606641	−0.303320	−	0.952889i	$-0.598095\pi$
$139$	−994.154	−0.606641	−0.303320	+	0.952889i	$0.598095\pi$
$140$	25.4844	0.0153845
$141$	0	0
$142$	1318.49	0.779192
$143$	−517.637	−0.302706
$144$	0	0
$145$	124.096	0.0710730
$146$	493.843	0.279937
$147$	0	0
$148$	−165.850	−0.0921134
$149$	−3284.09	−1.80566	−0.902829	−	0.429999i	$-0.858514\pi$
$149$	−3284.09	−1.80566	−0.902829	+	0.429999i	$0.858514\pi$
$150$	0	0
$151$	−1951.06	−1.05149	−0.525746	−	0.850642i	$-0.676214\pi$
$151$	−1951.06	−1.05149	−0.525746	+	0.850642i	$0.676214\pi$
$152$	754.947	0.402857
$153$	0	0
$154$	63.3570	0.0331523
$155$	398.943	0.206735
$156$	0	0
$157$	−1162.01	−0.590692	−0.295346	−	0.955390i	$-0.595435\pi$
$157$	−1162.01	−0.590692	−0.295346	+	0.955390i	$0.595435\pi$
$158$	571.121	0.287569
$159$	0	0
$160$	245.562	0.121334
$161$	488.240	0.238998
$162$	0	0
$163$	−47.8990	−0.0230168	−0.0115084	−	0.999934i	$-0.503663\pi$
$163$	−47.8990	−0.0230168	−0.0115084	+	0.999934i	$0.503663\pi$
$164$	−1821.74	−0.867402
$165$	0	0
$166$	624.390	0.291940
$167$	2521.02	1.16816	0.584079	−	0.811697i	$-0.301456\pi$
$167$	2521.02	1.16816	0.584079	+	0.811697i	$0.301456\pi$
$168$	0	0
$169$	−396.546	−0.180494
$170$	116.293	0.0524661
$171$	0	0
$172$	−604.011	−0.267764
$173$	−1701.47	−0.747746	−0.373873	−	0.927480i	$-0.621970\pi$
$173$	−1701.47	−0.747746	−0.373873	+	0.927480i	$0.621970\pi$
$174$	0	0
$175$	421.563	0.182098
$176$	−170.261	−0.0729200
$177$	0	0
$178$	−622.624	−0.262178
$179$	673.892	0.281391	0.140696	−	0.990053i	$-0.455066\pi$
$179$	673.892	0.281391	0.140696	+	0.990053i	$0.455066\pi$
$180$	0	0
$181$	−3639.03	−1.49440	−0.747202	−	0.664598i	$-0.768603\pi$
$181$	−3639.03	−1.49440	−0.747202	+	0.664598i	$0.768603\pi$
$182$	−220.369	−0.0897520
$183$	0	0
$184$	2969.62	1.18980
$185$	38.1365	0.0151559
$186$	0	0
$187$	−713.508	−0.279021
$188$	406.098	0.157541
$189$	0	0
$190$	−72.1755	−0.0275587
$191$	1587.62	0.601444	0.300722	−	0.953712i	$-0.402772\pi$
$191$	1587.62	0.601444	0.300722	+	0.953712i	$0.402772\pi$
$192$	0	0
$193$	−137.313	−0.0512126	−0.0256063	−	0.999672i	$-0.508152\pi$
$193$	−137.313	−0.0512126	−0.0256063	+	0.999672i	$0.508152\pi$
$194$	−571.910	−0.211653
$195$	0	0
$196$	1886.18	0.687382
$197$	1629.72	0.589403	0.294702	−	0.955589i	$-0.404780\pi$
$197$	1629.72	0.589403	0.294702	+	0.955589i	$0.404780\pi$
$198$	0	0
$199$	−3614.66	−1.28762	−0.643810	−	0.765185i	$-0.722647\pi$
$199$	−3614.66	−1.28762	−0.643810	+	0.765185i	$0.722647\pi$
$200$	2564.07	0.906534
$201$	0	0
$202$	2419.47	0.842739
$203$	−324.137	−0.112069
$204$	0	0
$205$	418.901	0.142719
$206$	−2608.28	−0.882171
$207$	0	0
$208$	592.206	0.197414
$209$	442.829	0.146560
$210$	0	0
$211$	−975.959	−0.318425	−0.159213	−	0.987244i	$-0.550896\pi$
$211$	−975.959	−0.318425	−0.159213	+	0.987244i	$0.550896\pi$
$212$	1051.25	0.340567
$213$	0	0
$214$	−1595.99	−0.509810
$215$	138.890	0.0440568
$216$	0	0
$217$	−1042.04	−0.325982
$218$	71.4590	0.0222010
$219$	0	0
$220$	90.9203	0.0278629
$221$	2481.73	0.755382
$222$	0	0
$223$	−33.7234	−0.0101268	−0.00506342	−	0.999987i	$-0.501612\pi$
$223$	−33.7234	−0.0101268	−0.00506342	+	0.999987i	$0.501612\pi$
$224$	−641.406	−0.191320
$225$	0	0
$226$	490.903	0.144488
$227$	−83.1752	−0.0243195	−0.0121598	−	0.999926i	$-0.503871\pi$
$227$	−83.1752	−0.0243195	−0.0121598	+	0.999926i	$0.503871\pi$
$228$	0	0
$229$	−2575.33	−0.743155	−0.371577	−	0.928402i	$-0.621183\pi$
$229$	−2575.33	−0.743155	−0.371577	+	0.928402i	$0.621183\pi$
$230$	−283.906	−0.0813921
$231$	0	0
$232$	−1971.49	−0.557909
$233$	110.403	0.0310419	0.0155209	−	0.999880i	$-0.495059\pi$
$233$	110.403	0.0310419	0.0155209	+	0.999880i	$0.495059\pi$
$234$	0	0
$235$	−93.3805	−0.0259212
$236$	3530.11	0.973689
$237$	0	0
$238$	−303.756	−0.0827292
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−6182.34	−1.65245	−0.826223	−	0.563343i	$-0.809515\pi$
$241$	−6182.34	−1.65245	−0.826223	+	0.563343i	$0.809515\pi$
$242$	−1795.54	−0.476949
$243$	0	0
$244$	3653.10	0.958467
$245$	−433.718	−0.113099
$246$	0	0
$247$	−1540.25	−0.396778
$248$	−6337.96	−1.62283
$249$	0	0
$250$	−493.675	−0.124891
$251$	6137.47	1.54340	0.771701	−	0.635985i	$-0.219406\pi$
$251$	6137.47	1.54340	0.771701	+	0.635985i	$0.219406\pi$
$252$	0	0
$253$	1741.89	0.432852
$254$	−2122.49	−0.524319
$255$	0	0
$256$	−3265.56	−0.797255
$257$	−5499.49	−1.33482	−0.667410	−	0.744690i	$-0.732597\pi$
$257$	−5499.49	−1.33482	−0.667410	+	0.744690i	$0.732597\pi$
$258$	0	0
$259$	−99.6122	−0.0238981
$260$	−316.241	−0.0754323
$261$	0	0
$262$	308.360	0.0727119
$263$	−5423.44	−1.27157	−0.635786	−	0.771865i	$-0.719324\pi$
$263$	−5423.44	−1.27157	−0.635786	+	0.771865i	$0.719324\pi$
$264$	0	0
$265$	−241.731	−0.0560355
$266$	188.522	0.0434549
$267$	0	0
$268$	−894.376	−0.203853
$269$	−3603.24	−0.816705	−0.408353	−	0.912824i	$-0.633897\pi$
$269$	−3603.24	−0.816705	−0.408353	+	0.912824i	$0.633897\pi$
$270$	0	0
$271$	3538.94	0.793267	0.396634	−	0.917977i	$-0.370178\pi$
$271$	3538.94	0.793267	0.396634	+	0.917977i	$0.370178\pi$
$272$	816.293	0.181967
$273$	0	0
$274$	−1817.06	−0.400630
$275$	1504.00	0.329799
$276$	0	0
$277$	−4110.37	−0.891582	−0.445791	−	0.895137i	$-0.647078\pi$
$277$	−4110.37	−0.891582	−0.445791	+	0.895137i	$0.647078\pi$
$278$	−1509.96	−0.325761
$279$	0	0
$280$	93.0976	0.0198702
$281$	−8282.33	−1.75830	−0.879150	−	0.476546i	$-0.841889\pi$
$281$	−8282.33	−1.75830	−0.879150	+	0.476546i	$0.841889\pi$
$282$	0	0
$283$	8256.34	1.73423	0.867117	−	0.498104i	$-0.165970\pi$
$283$	8256.34	1.73423	0.867117	+	0.498104i	$0.165970\pi$
$284$	−4942.15	−1.03261
$285$	0	0
$286$	−786.208	−0.162550
$287$	−1094.17	−0.225041
$288$	0	0
$289$	−1492.19	−0.303723
$290$	188.482	0.0381656
$291$	0	0
$292$	−1851.09	−0.370983
$293$	2777.20	0.553739	0.276870	−	0.960908i	$-0.410703\pi$
$293$	2777.20	0.553739	0.276870	+	0.960908i	$0.410703\pi$
$294$	0	0
$295$	−811.734	−0.160207
$296$	−605.870	−0.118971
$297$	0	0
$298$	−4988.01	−0.969623
$299$	−6058.66	−1.17184
$300$	0	0
$301$	−362.779	−0.0694692
$302$	−2963.35	−0.564642
$303$	0	0
$304$	−506.621	−0.0955813
$305$	−840.015	−0.157702
$306$	0	0
$307$	4019.16	0.747185	0.373592	−	0.927593i	$-0.378126\pi$
$307$	4019.16	0.747185	0.373592	+	0.927593i	$0.378126\pi$
$308$	−237.483	−0.0439346
$309$	0	0
$310$	605.931	0.111015
$311$	7494.28	1.36644	0.683218	−	0.730215i	$-0.260580\pi$
$311$	7494.28	1.36644	0.683218	+	0.730215i	$0.260580\pi$
$312$	0	0
$313$	612.382	0.110588	0.0552938	−	0.998470i	$-0.482390\pi$
$313$	612.382	0.110588	0.0552938	+	0.998470i	$0.482390\pi$
$314$	−1764.91	−0.317197
$315$	0	0
$316$	−2140.75	−0.381097
$317$	−4164.05	−0.737780	−0.368890	−	0.929473i	$-0.620262\pi$
$317$	−4164.05	−0.737780	−0.368890	+	0.929473i	$0.620262\pi$
$318$	0	0
$319$	−1156.42	−0.202969
$320$	226.803	0.0396208
$321$	0	0
$322$	741.559	0.128340
$323$	−2123.08	−0.365731
$324$	0	0
$325$	−5231.25	−0.892854
$326$	−72.7510	−0.0123598
$327$	0	0
$328$	−6655.03	−1.12031
$329$	243.909	0.0408728
$330$	0	0
$331$	4861.93	0.807359	0.403680	−	0.914900i	$-0.367731\pi$
$331$	4861.93	0.807359	0.403680	+	0.914900i	$0.367731\pi$
$332$	−2340.42	−0.386890
$333$	0	0
$334$	3829.03	0.627290
$335$	205.658	0.0335412
$336$	0	0
$337$	7850.07	1.26890	0.634452	−	0.772962i	$-0.281226\pi$
$337$	7850.07	1.26890	0.634452	+	0.772962i	$0.281226\pi$
$338$	−602.291	−0.0969239
$339$	0	0
$340$	−435.904	−0.0695301
$341$	−3717.66	−0.590388
$342$	0	0
$343$	2305.72	0.362965
$344$	−2206.53	−0.345837
$345$	0	0
$346$	−2584.26	−0.401533
$347$	−1414.73	−0.218867	−0.109433	−	0.993994i	$-0.534904\pi$
$347$	−1414.73	−0.218867	−0.109433	+	0.993994i	$0.534904\pi$
$348$	0	0
$349$	8636.33	1.32462	0.662309	−	0.749230i	$-0.269576\pi$
$349$	8636.33	1.32462	0.662309	+	0.749230i	$0.269576\pi$
$350$	640.287	0.0977851
$351$	0	0
$352$	−2288.33	−0.346502
$353$	11685.3	1.76188	0.880941	−	0.473226i	$-0.156910\pi$
$353$	11685.3	1.76188	0.880941	+	0.473226i	$0.156910\pi$
$354$	0	0
$355$	1136.43	0.169902
$356$	2333.80	0.347447
$357$	0	0
$358$	1023.53	0.151105
$359$	1126.22	0.165570	0.0827849	−	0.996567i	$-0.473619\pi$
$359$	1126.22	0.165570	0.0827849	+	0.996567i	$0.473619\pi$
$360$	0	0
$361$	−5541.34	−0.807893
$362$	−5527.10	−0.802481
$363$	0	0
$364$	826.018	0.118943
$365$	425.651	0.0610399
$366$	0	0
$367$	11129.1	1.58293	0.791463	−	0.611217i	$-0.209320\pi$
$367$	11129.1	1.58293	0.791463	+	0.611217i	$0.209320\pi$
$368$	−1992.82	−0.282290
$369$	0	0
$370$	57.9232	0.00813861
$371$	631.399	0.0883575
$372$	0	0
$373$	−10440.4	−1.44928	−0.724641	−	0.689126i	$-0.757994\pi$
$373$	−10440.4	−1.44928	−0.724641	+	0.689126i	$0.757994\pi$
$374$	−1083.70	−0.149832
$375$	0	0
$376$	1483.53	0.203476
$377$	4022.28	0.549490
$378$	0	0
$379$	−6149.13	−0.833403	−0.416701	−	0.909043i	$-0.636814\pi$
$379$	−6149.13	−0.833403	−0.416701	+	0.909043i	$0.636814\pi$
$380$	270.538	0.0365219
$381$	0	0
$382$	2411.33	0.322970
$383$	−150.613	−0.0200939	−0.0100470	−	0.999950i	$-0.503198\pi$
$383$	−150.613	−0.0200939	−0.0100470	+	0.999950i	$0.503198\pi$
$384$	0	0
$385$	54.6082	0.00722882
$386$	−208.557	−0.0275007
$387$	0	0
$388$	2143.71	0.280491
$389$	761.627	0.0992700	0.0496350	−	0.998767i	$-0.484194\pi$
$389$	761.627	0.0992700	0.0496350	+	0.998767i	$0.484194\pi$
$390$	0	0
$391$	−8351.22	−1.08015
$392$	6890.43	0.887804
$393$	0	0
$394$	2475.28	0.316504
$395$	492.257	0.0627041
$396$	0	0
$397$	−14041.0	−1.77506	−0.887529	−	0.460752i	$-0.847580\pi$
$397$	−14041.0	−1.77506	−0.887529	+	0.460752i	$0.847580\pi$
$398$	−5490.09	−0.691441
$399$	0	0
$400$	−1720.66	−0.215083
$401$	−4238.28	−0.527805	−0.263902	−	0.964549i	$-0.585010\pi$
$401$	−4238.28	−0.527805	−0.263902	+	0.964549i	$0.585010\pi$
$402$	0	0
$403$	12930.8	1.59834
$404$	−9068.99	−1.11683
$405$	0	0
$406$	−492.313	−0.0601800
$407$	−355.385	−0.0432820
$408$	0	0
$409$	−10802.1	−1.30594	−0.652970	−	0.757384i	$-0.726477\pi$
$409$	−10802.1	−1.30594	−0.652970	+	0.757384i	$0.726477\pi$
$410$	636.244	0.0766387
$411$	0	0
$412$	9776.70	1.16909
$413$	2120.24	0.252616
$414$	0	0
$415$	538.170	0.0636572
$416$	7959.32	0.938071
$417$	0	0
$418$	672.586	0.0787016
$419$	−3089.88	−0.360264	−0.180132	−	0.983642i	$-0.557652\pi$
$419$	−3089.88	−0.360264	−0.180132	+	0.983642i	$0.557652\pi$
$420$	0	0
$421$	4321.69	0.500300	0.250150	−	0.968207i	$-0.419520\pi$
$421$	4321.69	0.500300	0.250150	+	0.968207i	$0.419520\pi$
$422$	−1482.33	−0.170992
$423$	0	0
$424$	3840.35	0.439868
$425$	−7210.72	−0.822992
$426$	0	0
$427$	2194.11	0.248666
$428$	5982.30	0.675620
$429$	0	0
$430$	210.951	0.0236581
$431$	−10338.6	−1.15544	−0.577719	−	0.816235i	$-0.696057\pi$
$431$	−10338.6	−1.15544	−0.577719	+	0.816235i	$0.696057\pi$
$432$	0	0
$433$	−5044.22	−0.559837	−0.279919	−	0.960024i	$-0.590307\pi$
$433$	−5044.22	−0.559837	−0.279919	+	0.960024i	$0.590307\pi$
$434$	−1582.69	−0.175049
$435$	0	0
$436$	−267.852	−0.0294216
$437$	5183.08	0.567369
$438$	0	0
$439$	−9543.51	−1.03756	−0.518778	−	0.854909i	$-0.673613\pi$
$439$	−9543.51	−1.03756	−0.518778	+	0.854909i	$0.673613\pi$
$440$	332.143	0.0359870
$441$	0	0
$442$	3769.36	0.405634
$443$	−12762.2	−1.36874	−0.684369	−	0.729136i	$-0.739922\pi$
$443$	−12762.2	−1.36874	−0.684369	+	0.729136i	$0.739922\pi$
$444$	0	0
$445$	−536.648	−0.0571675
$446$	−51.2204	−0.00543802
$447$	0	0
$448$	−592.407	−0.0624746
$449$	6289.70	0.661090	0.330545	−	0.943790i	$-0.392767\pi$
$449$	6289.70	0.661090	0.330545	+	0.943790i	$0.392767\pi$
$450$	0	0
$451$	−3903.64	−0.407573
$452$	−1840.07	−0.191482
$453$	0	0
$454$	−126.330	−0.0130594
$455$	−189.939	−0.0195703
$456$	0	0
$457$	−2969.96	−0.304001	−0.152001	−	0.988380i	$-0.548572\pi$
$457$	−2969.96	−0.304001	−0.152001	+	0.988380i	$0.548572\pi$
$458$	−3911.51	−0.399067
$459$	0	0
$460$	1064.17	0.107864
$461$	−1378.37	−0.139256	−0.0696282	−	0.997573i	$-0.522181\pi$
$461$	−1378.37	−0.139256	−0.0696282	+	0.997573i	$0.522181\pi$
$462$	0	0
$463$	−18875.2	−1.89461	−0.947304	−	0.320336i	$-0.896204\pi$
$463$	−18875.2	−1.89461	−0.947304	+	0.320336i	$0.896204\pi$
$464$	1323.01	0.132369
$465$	0	0
$466$	167.685	0.0166692
$467$	2764.92	0.273973	0.136986	−	0.990573i	$-0.456258\pi$
$467$	2764.92	0.273973	0.136986	+	0.990573i	$0.456258\pi$
$468$	0	0
$469$	−537.177	−0.0528881
$470$	−141.830	−0.0139194
$471$	0	0
$472$	12895.9	1.25759
$473$	−1294.28	−0.125816
$474$	0	0
$475$	4475.24	0.432291
$476$	1138.58	0.109636
$477$	0	0
$478$	−363.003	−0.0347351
$479$	−10207.1	−0.973645	−0.486823	−	0.873501i	$-0.661844\pi$
$479$	−10207.1	−0.973645	−0.486823	+	0.873501i	$0.661844\pi$
$480$	0	0
$481$	1236.11	0.117176
$482$	−9389.99	−0.887349
$483$	0	0
$484$	6730.28	0.632070
$485$	−492.938	−0.0461508
$486$	0	0
$487$	−14408.4	−1.34067	−0.670334	−	0.742060i	$-0.733849\pi$
$487$	−14408.4	−1.34067	−0.670334	+	0.742060i	$0.733849\pi$
$488$	13345.2	1.23793
$489$	0	0
$490$	−658.749	−0.0607331
$491$	19289.3	1.77294	0.886472	−	0.462782i	$-0.153149\pi$
$491$	19289.3	1.77294	0.886472	+	0.462782i	$0.153149\pi$
$492$	0	0
$493$	5544.28	0.506495
$494$	−2339.40	−0.213066
$495$	0	0
$496$	4253.21	0.385029
$497$	−2968.33	−0.267903
$498$	0	0
$499$	−8649.30	−0.775944	−0.387972	−	0.921671i	$-0.626824\pi$
$499$	−8649.30	−0.775944	−0.387972	+	0.921671i	$0.626824\pi$
$500$	1850.46	0.165510
$501$	0	0
$502$	9321.84	0.828793
$503$	18146.8	1.60860	0.804301	−	0.594223i	$-0.202540\pi$
$503$	18146.8	1.60860	0.804301	+	0.594223i	$0.202540\pi$
$504$	0	0
$505$	2085.38	0.183758
$506$	2645.65	0.232438
$507$	0	0
$508$	7955.82	0.694847
$509$	9021.35	0.785588	0.392794	−	0.919626i	$-0.371508\pi$
$509$	9021.35	0.785588	0.392794	+	0.919626i	$0.371508\pi$
$510$	0	0
$511$	−1111.80	−0.0962485
$512$	4940.11	0.426414
$513$	0	0
$514$	−8352.85	−0.716786
$515$	−2248.11	−0.192356
$516$	0	0
$517$	870.191	0.0740251
$518$	−151.295	−0.0128331
$519$	0	0
$520$	−1155.26	−0.0974264
$521$	15170.7	1.27570	0.637851	−	0.770160i	$-0.279824\pi$
$521$	15170.7	1.27570	0.637851	+	0.770160i	$0.279824\pi$
$522$	0	0
$523$	9317.31	0.779001	0.389501	−	0.921026i	$-0.372648\pi$
$523$	9317.31	0.779001	0.389501	+	0.921026i	$0.372648\pi$
$524$	−1155.84	−0.0963605
$525$	0	0
$526$	−8237.33	−0.682823
$527$	17823.8	1.47327
$528$	0	0
$529$	8220.86	0.675669
$530$	−367.151	−0.0300906
$531$	0	0
$532$	−706.643	−0.0575881
$533$	13577.7	1.10341
$534$	0	0
$535$	−1375.60	−0.111164
$536$	−3267.26	−0.263292
$537$	0	0
$538$	−5472.75	−0.438563
$539$	4041.72	0.322985
$540$	0	0
$541$	1807.87	0.143672	0.0718360	−	0.997416i	$-0.477114\pi$
$541$	1807.87	0.143672	0.0718360	+	0.997416i	$0.477114\pi$
$542$	5375.09	0.425977
$543$	0	0
$544$	10971.1	0.864671
$545$	61.5915	0.00484090
$546$	0	0
$547$	−6407.32	−0.500836	−0.250418	−	0.968138i	$-0.580568\pi$
$547$	−6407.32	−0.500836	−0.250418	+	0.968138i	$0.580568\pi$
$548$	6810.95	0.530930
$549$	0	0
$550$	2284.34	0.177099
$551$	−3440.98	−0.266045
$552$	0	0
$553$	−1285.77	−0.0988726
$554$	−6243.00	−0.478772
$555$	0	0
$556$	5659.84	0.431710
$557$	−24204.3	−1.84123	−0.920617	−	0.390467i	$-0.872313\pi$
$557$	−24204.3	−1.84123	−0.920617	+	0.390467i	$0.872313\pi$
$558$	0	0
$559$	4501.79	0.340618
$560$	−62.4749	−0.00471437
$561$	0	0
$562$	−12579.5	−0.944191
$563$	−1207.42	−0.0903849	−0.0451925	−	0.998978i	$-0.514390\pi$
$563$	−1207.42	−0.0903849	−0.0451925	+	0.998978i	$0.514390\pi$
$564$	0	0
$565$	423.116	0.0315056
$566$	12540.1	0.931269
$567$	0	0
$568$	−18054.3	−1.33370
$569$	8015.99	0.590594	0.295297	−	0.955406i	$-0.404581\pi$
$569$	8015.99	0.590594	0.295297	+	0.955406i	$0.404581\pi$
$570$	0	0
$571$	5882.20	0.431107	0.215554	−	0.976492i	$-0.430844\pi$
$571$	5882.20	0.431107	0.215554	+	0.976492i	$0.430844\pi$
$572$	2946.97	0.215418
$573$	0	0
$574$	−1661.86	−0.120845
$575$	17603.6	1.27673
$576$	0	0
$577$	−4162.44	−0.300320	−0.150160	−	0.988662i	$-0.547979\pi$
$577$	−4162.44	−0.300320	−0.150160	+	0.988662i	$0.547979\pi$
$578$	−2266.40	−0.163097
$579$	0	0
$580$	−706.492	−0.0505785
$581$	−1405.70	−0.100375
$582$	0	0
$583$	2252.63	0.160025
$584$	−6762.26	−0.479151
$585$	0	0
$586$	4218.12	0.297353
$587$	10748.4	0.755766	0.377883	−	0.925853i	$-0.376652\pi$
$587$	10748.4	0.755766	0.377883	+	0.925853i	$0.376652\pi$
$588$	0	0
$589$	−11062.1	−0.773862
$590$	−1232.89	−0.0860296
$591$	0	0
$592$	406.580	0.0282269
$593$	8413.76	0.582651	0.291325	−	0.956624i	$-0.405904\pi$
$593$	8413.76	0.582651	0.291325	+	0.956624i	$0.405904\pi$
$594$	0	0
$595$	−261.811	−0.0180390
$596$	18696.7	1.28498
$597$	0	0
$598$	−9202.14	−0.629270
$599$	21331.3	1.45505	0.727523	−	0.686083i	$-0.240671\pi$
$599$	21331.3	1.45505	0.727523	+	0.686083i	$0.240671\pi$
$600$	0	0
$601$	−24780.6	−1.68190	−0.840951	−	0.541111i	$-0.818004\pi$
$601$	−24780.6	−1.68190	−0.840951	+	0.541111i	$0.818004\pi$
$602$	−551.003	−0.0373044
$603$	0	0
$604$	11107.6	0.748284
$605$	−1547.60	−0.103998
$606$	0	0
$607$	−1492.65	−0.0998099	−0.0499050	−	0.998754i	$-0.515892\pi$
$607$	−1492.65	−0.0998099	−0.0499050	+	0.998754i	$0.515892\pi$
$608$	−6809.05	−0.454183
$609$	0	0
$610$	−1275.85	−0.0846846
$611$	−3026.71	−0.200405
$612$	0	0
$613$	−15960.8	−1.05163	−0.525815	−	0.850599i	$-0.676240\pi$
$613$	−15960.8	−1.05163	−0.525815	+	0.850599i	$0.676240\pi$
$614$	6104.47	0.401232
$615$	0	0
$616$	−867.555	−0.0567448
$617$	−25115.6	−1.63876	−0.819382	−	0.573248i	$-0.805683\pi$
$617$	−25115.6	−1.63876	−0.819382	+	0.573248i	$0.805683\pi$
$618$	0	0
$619$	−15306.3	−0.993880	−0.496940	−	0.867785i	$-0.665543\pi$
$619$	−15306.3	−0.993880	−0.496940	+	0.867785i	$0.665543\pi$
$620$	−2271.23	−0.147121
$621$	0	0
$622$	11382.6	0.733764
$623$	1401.72	0.0901424
$624$	0	0
$625$	14985.3	0.959058
$626$	930.111	0.0593845
$627$	0	0
$628$	6615.48	0.420361
$629$	1703.84	0.108007
$630$	0	0
$631$	−13659.8	−0.861790	−0.430895	−	0.902402i	$-0.641802\pi$
$631$	−13659.8	−0.861790	−0.430895	+	0.902402i	$0.641802\pi$
$632$	−7820.43	−0.492215
$633$	0	0
$634$	−6324.52	−0.396181
$635$	−1829.41	−0.114327
$636$	0	0
$637$	−14058.0	−0.874407
$638$	−1756.42	−0.108992
$639$	0	0
$640$	−1620.02	−0.100058
$641$	−9265.02	−0.570899	−0.285450	−	0.958394i	$-0.592143\pi$
$641$	−9265.02	−0.570899	−0.285450	+	0.958394i	$0.592143\pi$
$642$	0	0
$643$	23119.8	1.41797	0.708987	−	0.705222i	$-0.249153\pi$
$643$	23119.8	1.41797	0.708987	+	0.705222i	$0.249153\pi$
$644$	−2779.61	−0.170081
$645$	0	0
$646$	−3224.62	−0.196394
$647$	18701.5	1.13637	0.568187	−	0.822900i	$-0.307645\pi$
$647$	18701.5	1.13637	0.568187	+	0.822900i	$0.307645\pi$
$648$	0	0
$649$	7564.35	0.457514
$650$	−7945.43	−0.479455
$651$	0	0
$652$	272.695	0.0163797
$653$	−8503.50	−0.509598	−0.254799	−	0.966994i	$-0.582009\pi$
$653$	−8503.50	−0.509598	−0.254799	+	0.966994i	$0.582009\pi$
$654$	0	0
$655$	265.780	0.0158548
$656$	4465.99	0.265804
$657$	0	0
$658$	370.459	0.0219483
$659$	−7044.42	−0.416406	−0.208203	−	0.978086i	$-0.566762\pi$
$659$	−7044.42	−0.416406	−0.208203	+	0.978086i	$0.566762\pi$
$660$	0	0
$661$	31948.6	1.87997	0.939983	−	0.341220i	$-0.110840\pi$
$661$	31948.6	1.87997	0.939983	+	0.341220i	$0.110840\pi$
$662$	7384.50	0.433545
$663$	0	0
$664$	−8549.85	−0.499696
$665$	162.490	0.00947530
$666$	0	0
$667$	−13535.3	−0.785738
$668$	−14352.5	−0.831309
$669$	0	0
$670$	312.362	0.0180113
$671$	7827.90	0.450362
$672$	0	0
$673$	−11639.4	−0.666667	−0.333334	−	0.942809i	$-0.608174\pi$
$673$	−11639.4	−0.666667	−0.333334	+	0.942809i	$0.608174\pi$
$674$	11923.0	0.681390
$675$	0	0
$676$	2257.59	0.128447
$677$	−20961.3	−1.18997	−0.594984	−	0.803738i	$-0.702842\pi$
$677$	−20961.3	−1.18997	−0.594984	+	0.803738i	$0.702842\pi$
$678$	0	0
$679$	1287.55	0.0727711
$680$	−1592.41	−0.0898031
$681$	0	0
$682$	−5646.52	−0.317033
$683$	4132.83	0.231535	0.115767	−	0.993276i	$-0.463067\pi$
$683$	4132.83	0.231535	0.115767	+	0.993276i	$0.463067\pi$
$684$	0	0
$685$	−1566.15	−0.0873569
$686$	3502.02	0.194909
$687$	0	0
$688$	1480.73	0.0820528
$689$	−7835.15	−0.433230
$690$	0	0
$691$	26357.0	1.45104	0.725518	−	0.688203i	$-0.241600\pi$
$691$	26357.0	1.45104	0.725518	+	0.688203i	$0.241600\pi$
$692$	9686.66	0.532127
$693$	0	0
$694$	−2148.75	−0.117529
$695$	−1301.46	−0.0710318
$696$	0	0
$697$	18715.4	1.01707
$698$	13117.2	0.711309
$699$	0	0
$700$	−2400.01	−0.129588
$701$	1377.72	0.0742306	0.0371153	−	0.999311i	$-0.488183\pi$
$701$	1377.72	0.0742306	0.0371153	+	0.999311i	$0.488183\pi$
$702$	0	0
$703$	−1057.47	−0.0567327
$704$	−2113.52	−0.113148
$705$	0	0
$706$	17748.1	0.946115
$707$	−5446.99	−0.289752
$708$	0	0
$709$	18861.9	0.999114	0.499557	−	0.866281i	$-0.333496\pi$
$709$	18861.9	0.999114	0.499557	+	0.866281i	$0.333496\pi$
$710$	1726.05	0.0912358
$711$	0	0
$712$	8525.66	0.448754
$713$	−43513.2	−2.28553
$714$	0	0
$715$	−677.643	−0.0354440
$716$	−3836.55	−0.200250
$717$	0	0
$718$	1710.55	0.0889095
$719$	9489.44	0.492206	0.246103	−	0.969244i	$-0.420850\pi$
$719$	9489.44	0.492206	0.246103	+	0.969244i	$0.420850\pi$
$720$	0	0
$721$	5872.05	0.303310
$722$	−8416.41	−0.433832
$723$	0	0
$724$	20717.4	1.06348
$725$	−11686.8	−0.598671
$726$	0	0
$727$	23738.8	1.21104	0.605519	−	0.795831i	$-0.292965\pi$
$727$	23738.8	1.21104	0.605519	+	0.795831i	$0.292965\pi$
$728$	3017.54	0.153623
$729$	0	0
$730$	646.495	0.0327779
$731$	6205.24	0.313966
$732$	0	0
$733$	20537.0	1.03486	0.517429	−	0.855726i	$-0.326889\pi$
$733$	20537.0	1.03486	0.517429	+	0.855726i	$0.326889\pi$
$734$	16903.3	0.850018
$735$	0	0
$736$	−26783.7	−1.34139
$737$	−1916.48	−0.0957861
$738$	0	0
$739$	11967.2	0.595699	0.297849	−	0.954613i	$-0.403731\pi$
$739$	11967.2	0.595699	0.297849	+	0.954613i	$0.403731\pi$
$740$	−217.116	−0.0107856
$741$	0	0
$742$	958.995	0.0474472
$743$	−9680.80	−0.478000	−0.239000	−	0.971020i	$-0.576820\pi$
$743$	−9680.80	−0.478000	−0.239000	+	0.971020i	$0.576820\pi$
$744$	0	0
$745$	−4299.23	−0.211425
$746$	−15857.3	−0.778252
$747$	0	0
$748$	4062.09	0.198562
$749$	3593.07	0.175284
$750$	0	0
$751$	−32948.6	−1.60095	−0.800474	−	0.599368i	$-0.795419\pi$
$751$	−32948.6	−1.60095	−0.800474	+	0.599368i	$0.795419\pi$
$752$	−995.547	−0.0482764
$753$	0	0
$754$	6109.19	0.295071
$755$	−2554.15	−0.123119
$756$	0	0
$757$	−25971.5	−1.24696	−0.623482	−	0.781838i	$-0.714282\pi$
$757$	−25971.5	−1.24696	−0.623482	+	0.781838i	$0.714282\pi$
$758$	−9339.55	−0.447530
$759$	0	0
$760$	988.308	0.0471707
$761$	41224.7	1.96373	0.981863	−	0.189590i	$-0.0607160\pi$
$761$	41224.7	1.96373	0.981863	+	0.189590i	$0.0607160\pi$
$762$	0	0
$763$	−160.877	−0.00763319
$764$	−9038.50	−0.428012
$765$	0	0
$766$	−228.757	−0.0107903
$767$	−26310.5	−1.23861
$768$	0	0
$769$	29537.1	1.38509	0.692547	−	0.721373i	$-0.256489\pi$
$769$	29537.1	1.38509	0.692547	+	0.721373i	$0.256489\pi$
$770$	82.9412	0.00388181
$771$	0	0
$772$	781.742	0.0364450
$773$	38484.6	1.79068	0.895341	−	0.445382i	$-0.146932\pi$
$773$	38484.6	1.79068	0.895341	+	0.445382i	$0.146932\pi$
$774$	0	0
$775$	−37570.7	−1.74139
$776$	7831.24	0.362275
$777$	0	0
$778$	1156.79	0.0533071
$779$	−11615.5	−0.534234
$780$	0	0
$781$	−10590.1	−0.485202
$782$	−12684.2	−0.580032
$783$	0	0
$784$	−4623.95	−0.210639
$785$	−1521.20	−0.0691643
$786$	0	0
$787$	−29099.8	−1.31804	−0.659019	−	0.752126i	$-0.729028\pi$
$787$	−29099.8	−1.31804	−0.659019	+	0.752126i	$0.729028\pi$
$788$	−9278.18	−0.419443
$789$	0	0
$790$	747.659	0.0336716
$791$	−1105.18	−0.0496783
$792$	0	0
$793$	−27227.1	−1.21925
$794$	−21326.0	−0.953190
$795$	0	0
$796$	20578.7	0.916323
$797$	34241.4	1.52182	0.760912	−	0.648856i	$-0.224752\pi$
$797$	34241.4	1.52182	0.760912	+	0.648856i	$0.224752\pi$
$798$	0	0
$799$	−4172.00	−0.184724
$800$	−23125.9	−1.02203
$801$	0	0
$802$	−6437.27	−0.283426
$803$	−3966.54	−0.174316
$804$	0	0
$805$	639.160	0.0279844
$806$	19639.8	0.858292
$807$	0	0
$808$	−33130.1	−1.44247
$809$	7123.31	0.309570	0.154785	−	0.987948i	$-0.450531\pi$
$809$	7123.31	0.309570	0.154785	+	0.987948i	$0.450531\pi$
$810$	0	0
$811$	6068.94	0.262774	0.131387	−	0.991331i	$-0.458057\pi$
$811$	6068.94	0.262774	0.131387	+	0.991331i	$0.458057\pi$
$812$	1845.35	0.0797527
$813$	0	0
$814$	−539.773	−0.0232420
$815$	−62.7051	−0.00269505
$816$	0	0
$817$	−3851.20	−0.164916
$818$	−16406.7	−0.701278
$819$	0	0
$820$	−2384.86	−0.101564
$821$	3672.62	0.156121	0.0780605	−	0.996949i	$-0.475127\pi$
$821$	3672.62	0.156121	0.0780605	+	0.996949i	$0.475127\pi$
$822$	0	0
$823$	19239.0	0.814860	0.407430	−	0.913237i	$-0.366425\pi$
$823$	19239.0	0.814860	0.407430	+	0.913237i	$0.366425\pi$
$824$	35715.5	1.50996
$825$	0	0
$826$	3220.31	0.135652
$827$	−11069.8	−0.465457	−0.232729	−	0.972542i	$-0.574765\pi$
$827$	−11069.8	−0.465457	−0.232729	+	0.972542i	$0.574765\pi$
$828$	0	0
$829$	−8279.23	−0.346863	−0.173432	−	0.984846i	$-0.555486\pi$
$829$	−8279.23	−0.346863	−0.173432	+	0.984846i	$0.555486\pi$
$830$	817.395	0.0341834
$831$	0	0
$832$	7351.29	0.306322
$833$	−19377.4	−0.805988
$834$	0	0
$835$	3300.29	0.136780
$836$	−2521.08	−0.104298
$837$	0	0
$838$	−4693.03	−0.193458
$839$	−4140.85	−0.170391	−0.0851954	−	0.996364i	$-0.527151\pi$
$839$	−4140.85	−0.170391	−0.0851954	+	0.996364i	$0.527151\pi$
$840$	0	0
$841$	−15403.1	−0.631559
$842$	6563.96	0.268657
$843$	0	0
$844$	5556.26	0.226604
$845$	−519.123	−0.0211342
$846$	0	0
$847$	4042.32	0.163986
$848$	−2577.14	−0.104362
$849$	0	0
$850$	−10951.9	−0.441939
$851$	−4159.59	−0.167554
$852$	0	0
$853$	18728.2	0.751749	0.375874	−	0.926671i	$-0.377342\pi$
$853$	18728.2	0.751749	0.375874	+	0.926671i	$0.377342\pi$
$854$	3332.51	0.133532
$855$	0	0
$856$	21854.0	0.872612
$857$	39699.5	1.58239	0.791196	−	0.611563i	$-0.209459\pi$
$857$	39699.5	1.58239	0.791196	+	0.611563i	$0.209459\pi$
$858$	0	0
$859$	5270.13	0.209330	0.104665	−	0.994508i	$-0.466623\pi$
$859$	5270.13	0.209330	0.104665	+	0.994508i	$0.466623\pi$
$860$	−790.717	−0.0313526
$861$	0	0
$862$	−15702.7	−0.620460
$863$	−5645.98	−0.222702	−0.111351	−	0.993781i	$-0.535518\pi$
$863$	−5645.98	−0.222702	−0.111351	+	0.993781i	$0.535518\pi$
$864$	0	0
$865$	−2227.41	−0.0875539
$866$	−7661.36	−0.300628
$867$	0	0
$868$	5932.44	0.231982
$869$	−4587.23	−0.179069
$870$	0	0
$871$	6665.93	0.259318
$872$	−978.497	−0.0380001
$873$	0	0
$874$	7872.26	0.304672
$875$	1111.42	0.0429402
$876$	0	0
$877$	41839.6	1.61097	0.805487	−	0.592614i	$-0.201904\pi$
$877$	41839.6	1.61097	0.805487	+	0.592614i	$0.201904\pi$
$878$	−14495.1	−0.557158
$879$	0	0
$880$	−222.891	−0.00853823
$881$	−33916.9	−1.29704	−0.648518	−	0.761199i	$-0.724611\pi$
$881$	−33916.9	−1.29704	−0.648518	+	0.761199i	$0.724611\pi$
$882$	0	0
$883$	48638.9	1.85371	0.926857	−	0.375414i	$-0.122500\pi$
$883$	48638.9	1.85371	0.926857	+	0.375414i	$0.122500\pi$
$884$	−14128.8	−0.537561
$885$	0	0
$886$	−19383.8	−0.735000
$887$	26256.9	0.993937	0.496968	−	0.867769i	$-0.334446\pi$
$887$	26256.9	0.993937	0.496968	+	0.867769i	$0.334446\pi$
$888$	0	0
$889$	4778.39	0.180272
$890$	−815.082	−0.0306985
$891$	0	0
$892$	191.991	0.00720667
$893$	2589.30	0.0970297
$894$	0	0
$895$	882.199	0.0329482
$896$	4231.47	0.157772
$897$	0	0
$898$	9553.06	0.355000
$899$	28887.9	1.07171
$900$	0	0
$901$	−10799.9	−0.399331
$902$	−5929.01	−0.218863
$903$	0	0
$904$	−6722.00	−0.247312
$905$	−4763.89	−0.174980
$906$	0	0
$907$	−32099.4	−1.17513	−0.587565	−	0.809177i	$-0.699914\pi$
$907$	−32099.4	−1.17513	−0.587565	+	0.809177i	$0.699914\pi$
$908$	473.527	0.0173068
$909$	0	0
$910$	−288.487	−0.0105091
$911$	−26948.3	−0.980064	−0.490032	−	0.871704i	$-0.663015\pi$
$911$	−26948.3	−0.980064	−0.490032	+	0.871704i	$0.663015\pi$
$912$	0	0
$913$	−5015.08	−0.181791
$914$	−4510.89	−0.163246
$915$	0	0
$916$	14661.7	0.528859
$917$	−694.214	−0.0250000
$918$	0	0
$919$	29275.2	1.05082	0.525409	−	0.850850i	$-0.323912\pi$
$919$	29275.2	1.05082	0.525409	+	0.850850i	$0.323912\pi$
$920$	3887.55	0.139314
$921$	0	0
$922$	−2093.53	−0.0747795
$923$	36834.6	1.31357
$924$	0	0
$925$	−3591.53	−0.127663
$926$	−28668.4	−1.01739
$927$	0	0
$928$	17781.4	0.628990
$929$	43112.1	1.52256	0.761282	−	0.648421i	$-0.224570\pi$
$929$	43112.1	1.52256	0.761282	+	0.648421i	$0.224570\pi$
$930$	0	0
$931$	12026.3	0.423359
$932$	−628.540	−0.0220907
$933$	0	0
$934$	4199.47	0.147121
$935$	−934.060	−0.0326706
$936$	0	0
$937$	−34955.5	−1.21873	−0.609363	−	0.792891i	$-0.708575\pi$
$937$	−34955.5	−1.21873	−0.609363	+	0.792891i	$0.708575\pi$
$938$	−815.886	−0.0284005
$939$	0	0
$940$	531.627	0.0184466
$941$	3457.37	0.119774	0.0598868	−	0.998205i	$-0.480926\pi$
$941$	3457.37	0.119774	0.0598868	+	0.998205i	$0.480926\pi$
$942$	0	0
$943$	−45690.0	−1.57781
$944$	−8654.05	−0.298374
$945$	0	0
$946$	−1965.81	−0.0675622
$947$	−44902.2	−1.54079	−0.770393	−	0.637569i	$-0.779940\pi$
$947$	−44902.2	−1.54079	−0.770393	+	0.637569i	$0.779940\pi$
$948$	0	0
$949$	13796.5	0.471921
$950$	6797.17	0.232136
$951$	0	0
$952$	4159.36	0.141603
$953$	10911.0	0.370873	0.185436	−	0.982656i	$-0.440630\pi$
$953$	10911.0	0.370873	0.185436	+	0.982656i	$0.440630\pi$
$954$	0	0
$955$	2078.36	0.0704233
$956$	1360.66	0.0460322
$957$	0	0
$958$	−15503.0	−0.522839
$959$	4090.77	0.137745
$960$	0	0
$961$	63077.8	2.11734
$962$	1877.45	0.0629224
$963$	0	0
$964$	35196.8	1.17595
$965$	−179.758	−0.00599650
$966$	0	0
$967$	6865.08	0.228300	0.114150	−	0.993464i	$-0.463586\pi$
$967$	6865.08	0.228300	0.114150	+	0.993464i	$0.463586\pi$
$968$	24586.5	0.816365
$969$	0	0
$970$	−748.693	−0.0247826
$971$	−42271.3	−1.39706	−0.698532	−	0.715579i	$-0.746163\pi$
$971$	−42271.3	−1.39706	−0.698532	+	0.715579i	$0.746163\pi$
$972$	0	0
$973$	3399.40	0.112004
$974$	−21884.0	−0.719927
$975$	0	0
$976$	−8955.56	−0.293709
$977$	40446.3	1.32445	0.662227	−	0.749303i	$-0.269611\pi$
$977$	40446.3	1.32445	0.662227	+	0.749303i	$0.269611\pi$
$978$	0	0
$979$	5000.89	0.163258
$980$	2469.21	0.0804858
$981$	0	0
$982$	29297.4	0.952055
$983$	−14617.2	−0.474280	−0.237140	−	0.971476i	$-0.576210\pi$
$983$	−14617.2	−0.474280	−0.237140	+	0.971476i	$0.576210\pi$
$984$	0	0
$985$	2133.48	0.0690134
$986$	8420.88	0.271983
$987$	0	0
$988$	8768.86	0.282363
$989$	−15148.9	−0.487064
$990$	0	0
$991$	25438.8	0.815430	0.407715	−	0.913109i	$-0.366326\pi$
$991$	25438.8	0.815430	0.407715	+	0.913109i	$0.366326\pi$
$992$	57163.6	1.82958
$993$	0	0
$994$	−4508.43	−0.143862
$995$	−4731.99	−0.150768
$996$	0	0
$997$	−41464.6	−1.31715	−0.658574	−	0.752516i	$-0.728840\pi$
$997$	−41464.6	−1.31715	−0.658574	+	0.752516i	$0.728840\pi$
$998$	−13136.9	−0.416675
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.a.1.15		22
3.2	odd	2		239.4.a.a.1.8	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.a.1.8	✓	22	3.2	odd	2
2151.4.a.a.1.15		22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.51884 q^{2} -5.69312 q^{4} +1.30911 q^{5} -3.41938 q^{7} -20.7977 q^{8} +O(q^{10})\)	\(q+1.51884 q^{2} -5.69312 q^{4} +1.30911 q^{5} -3.41938 q^{7} -20.7977 q^{8} +1.98833 q^{10} -12.1993 q^{11} +42.4318 q^{13} -5.19350 q^{14} +13.9567 q^{16} +58.4877 q^{17} -36.2996 q^{19} -7.45292 q^{20} -18.5288 q^{22} -142.786 q^{23} -123.286 q^{25} +64.4470 q^{26} +19.4670 q^{28} +94.7940 q^{29} +304.744 q^{31} +187.579 q^{32} +88.8334 q^{34} -4.47635 q^{35} +29.1316 q^{37} -55.1333 q^{38} -27.2264 q^{40} +319.989 q^{41} +106.095 q^{43} +69.4520 q^{44} -216.869 q^{46} -71.3313 q^{47} -331.308 q^{49} -187.252 q^{50} -241.569 q^{52} -184.653 q^{53} -15.9702 q^{55} +71.1152 q^{56} +143.977 q^{58} -620.066 q^{59} -641.669 q^{61} +462.857 q^{62} +173.250 q^{64} +55.5478 q^{65} +157.098 q^{67} -332.978 q^{68} -6.79886 q^{70} +868.090 q^{71} +325.145 q^{73} +44.2463 q^{74} +206.658 q^{76} +41.7140 q^{77} +376.024 q^{79} +18.2708 q^{80} +486.013 q^{82} +411.096 q^{83} +76.5668 q^{85} +161.141 q^{86} +253.717 q^{88} -409.934 q^{89} -145.090 q^{91} +812.899 q^{92} -108.341 q^{94} -47.5201 q^{95} -376.544 q^{97} -503.204 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Introduction

L-functions

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