LMFDB - Embedded newform 2178.4.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,4,Mod(1,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, 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("2178.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2178.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:	$128.506159993$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.978025.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 $ x^4 - 2x^3 - 99x^2 + 100*x + 2420
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.19378$ of defining polynomial
Character	$\chi$	$=$	2178.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -14.9181 q^{5} -21.7679 q^{7} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} -14.9181 q^{5} -21.7679 q^{7} -8.00000 q^{8} +29.8363 q^{10} +44.0286 q^{13} +43.5357 q^{14} +16.0000 q^{16} -24.9145 q^{17} -21.9573 q^{19} -59.6725 q^{20} -177.749 q^{23} +97.5508 q^{25} -88.0571 q^{26} -87.0714 q^{28} -149.396 q^{29} +75.1436 q^{31} -32.0000 q^{32} +49.8290 q^{34} +324.736 q^{35} +222.336 q^{37} +43.9145 q^{38} +119.345 q^{40} -253.121 q^{41} -130.623 q^{43} +355.498 q^{46} -499.093 q^{47} +130.839 q^{49} -195.102 q^{50} +176.114 q^{52} -12.9421 q^{53} +174.143 q^{56} +298.793 q^{58} -35.5614 q^{59} -538.343 q^{61} -150.287 q^{62} +64.0000 q^{64} -656.824 q^{65} -519.621 q^{67} -99.6580 q^{68} -649.472 q^{70} -78.4486 q^{71} -1144.07 q^{73} -444.673 q^{74} -87.8290 q^{76} -772.546 q^{79} -238.690 q^{80} +506.243 q^{82} +537.242 q^{83} +371.678 q^{85} +261.247 q^{86} -667.089 q^{89} -958.407 q^{91} -710.996 q^{92} +998.187 q^{94} +327.561 q^{95} -179.654 q^{97} -261.679 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} - 25 q^{5} + 3 q^{7} - 32 q^{8}+O(q^{10}) $ 4 * q - 8 * q^2 + 16 * q^4 - 25 * q^5 + 3 * q^7 - 32 * q^8	$ 4 q - 8 q^{2} + 16 q^{4} - 25 q^{5} + 3 q^{7} - 32 q^{8} + 50 q^{10} - 41 q^{13} - 6 q^{14} + 64 q^{16} - 52 q^{17} + 16 q^{19} - 100 q^{20} - 314 q^{23} - 21 q^{25} + 82 q^{26} + 12 q^{28} - 561 q^{29} + 199 q^{31} - 128 q^{32} + 104 q^{34} + 714 q^{35} + 357 q^{37} - 32 q^{38} + 200 q^{40} - 32 q^{41} - 721 q^{43} + 628 q^{46} - 403 q^{47} + 823 q^{49} + 42 q^{50} - 164 q^{52} + 133 q^{53} - 24 q^{56} + 1122 q^{58} - 1016 q^{59} - 919 q^{61} - 398 q^{62} + 256 q^{64} - 69 q^{65} + 289 q^{67} - 208 q^{68} - 1428 q^{70} + 1205 q^{71} - 1234 q^{73} - 714 q^{74} + 64 q^{76} - 603 q^{79} - 400 q^{80} + 64 q^{82} - 1514 q^{83} + 717 q^{85} + 1442 q^{86} + 1101 q^{89} - 2306 q^{91} - 1256 q^{92} + 806 q^{94} + 1766 q^{95} + 2116 q^{97} - 1646 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 - 25 * q^5 + 3 * q^7 - 32 * q^8 + 50 * q^10 - 41 * q^13 - 6 * q^14 + 64 * q^16 - 52 * q^17 + 16 * q^19 - 100 * q^20 - 314 * q^23 - 21 * q^25 + 82 * q^26 + 12 * q^28 - 561 * q^29 + 199 * q^31 - 128 * q^32 + 104 * q^34 + 714 * q^35 + 357 * q^37 - 32 * q^38 + 200 * q^40 - 32 * q^41 - 721 * q^43 + 628 * q^46 - 403 * q^47 + 823 * q^49 + 42 * q^50 - 164 * q^52 + 133 * q^53 - 24 * q^56 + 1122 * q^58 - 1016 * q^59 - 919 * q^61 - 398 * q^62 + 256 * q^64 - 69 * q^65 + 289 * q^67 - 208 * q^68 - 1428 * q^70 + 1205 * q^71 - 1234 * q^73 - 714 * q^74 + 64 * q^76 - 603 * q^79 - 400 * q^80 + 64 * q^82 - 1514 * q^83 + 717 * q^85 + 1442 * q^86 + 1101 * q^89 - 2306 * q^91 - 1256 * q^92 + 806 * q^94 + 1766 * q^95 + 2116 * q^97 - 1646 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−14.9181	−1.33432	−0.667159	−	0.744915i	$-0.732490\pi$
$5$	−14.9181	−1.33432	−0.667159	+	0.744915i	$0.732490\pi$
$6$	0	0
$7$	−21.7679	−1.17535	−0.587677	−	0.809096i	$-0.699957\pi$
$7$	−21.7679	−1.17535	−0.587677	+	0.809096i	$0.699957\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	29.8363	0.943506
$11$	0	0
$11$	0	0
$12$	0	0
$13$	44.0286	0.939333	0.469666	−	0.882844i	$-0.344374\pi$
$13$	44.0286	0.939333	0.469666	+	0.882844i	$0.344374\pi$
$14$	43.5357	0.831101
$15$	0	0
$16$	16.0000	0.250000
$17$	−24.9145	−0.355450	−0.177725	−	0.984080i	$-0.556874\pi$
$17$	−24.9145	−0.355450	−0.177725	+	0.984080i	$0.556874\pi$
$18$	0	0
$19$	−21.9573	−0.265123	−0.132562	−	0.991175i	$-0.542320\pi$
$19$	−21.9573	−0.265123	−0.132562	+	0.991175i	$0.542320\pi$
$20$	−59.6725	−0.667159
$21$	0	0
$22$	0	0
$23$	−177.749	−1.61145	−0.805723	−	0.592293i	$-0.798223\pi$
$23$	−177.749	−1.61145	−0.805723	+	0.592293i	$0.798223\pi$
$24$	0	0
$25$	97.5508	0.780407
$26$	−88.0571	−0.664208
$27$	0	0
$28$	−87.0714	−0.587677
$29$	−149.396	−0.956628	−0.478314	−	0.878189i	$-0.658752\pi$
$29$	−149.396	−0.956628	−0.478314	+	0.878189i	$0.658752\pi$
$30$	0	0
$31$	75.1436	0.435361	0.217681	−	0.976020i	$-0.430151\pi$
$31$	75.1436	0.435361	0.217681	+	0.976020i	$0.430151\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	49.8290	0.251341
$35$	324.736	1.56830
$36$	0	0
$37$	222.336	0.987889	0.493944	−	0.869493i	$-0.335555\pi$
$37$	222.336	0.987889	0.493944	+	0.869493i	$0.335555\pi$
$38$	43.9145	0.187470
$39$	0	0
$40$	119.345	0.471753
$41$	−253.121	−0.964168	−0.482084	−	0.876125i	$-0.660120\pi$
$41$	−253.121	−0.964168	−0.482084	+	0.876125i	$0.660120\pi$
$42$	0	0
$43$	−130.623	−0.463253	−0.231626	−	0.972805i	$-0.574405\pi$
$43$	−130.623	−0.463253	−0.231626	+	0.972805i	$0.574405\pi$
$44$	0	0
$45$	0	0
$46$	355.498	1.13946
$47$	−499.093	−1.54894	−0.774471	−	0.632610i	$-0.781984\pi$
$47$	−499.093	−1.54894	−0.774471	+	0.632610i	$0.781984\pi$
$48$	0	0
$49$	130.839	0.381456
$50$	−195.102	−0.551831
$51$	0	0
$52$	176.114	0.469666
$53$	−12.9421	−0.0335422	−0.0167711	−	0.999859i	$-0.505339\pi$
$53$	−12.9421	−0.0335422	−0.0167711	+	0.999859i	$0.505339\pi$
$54$	0	0
$55$	0	0
$56$	174.143	0.415550
$57$	0	0
$58$	298.793	0.676438
$59$	−35.5614	−0.0784694	−0.0392347	−	0.999230i	$-0.512492\pi$
$59$	−35.5614	−0.0784694	−0.0392347	+	0.999230i	$0.512492\pi$
$60$	0	0
$61$	−538.343	−1.12996	−0.564982	−	0.825103i	$-0.691117\pi$
$61$	−538.343	−1.12996	−0.564982	+	0.825103i	$0.691117\pi$
$62$	−150.287	−0.307847
$63$	0	0
$64$	64.0000	0.125000
$65$	−656.824	−1.25337
$66$	0	0
$67$	−519.621	−0.947491	−0.473745	−	0.880662i	$-0.657098\pi$
$67$	−519.621	−0.947491	−0.473745	+	0.880662i	$0.657098\pi$
$68$	−99.6580	−0.177725
$69$	0	0
$70$	−649.472	−1.10895
$71$	−78.4486	−0.131129	−0.0655644	−	0.997848i	$-0.520885\pi$
$71$	−78.4486	−0.131129	−0.0655644	+	0.997848i	$0.520885\pi$
$72$	0	0
$73$	−1144.07	−1.83429	−0.917145	−	0.398554i	$-0.869512\pi$
$73$	−1144.07	−1.83429	−0.917145	+	0.398554i	$0.869512\pi$
$74$	−444.673	−0.698543
$75$	0	0
$76$	−87.8290	−0.132562
$77$	0	0
$78$	0	0
$79$	−772.546	−1.10023	−0.550115	−	0.835089i	$-0.685416\pi$
$79$	−772.546	−1.10023	−0.550115	+	0.835089i	$0.685416\pi$
$80$	−238.690	−0.333580
$81$	0	0
$82$	506.243	0.681770
$83$	537.242	0.710481	0.355241	−	0.934775i	$-0.384399\pi$
$83$	537.242	0.710481	0.355241	+	0.934775i	$0.384399\pi$
$84$	0	0
$85$	371.678	0.474284
$86$	261.247	0.327569
$87$	0	0
$88$	0	0
$89$	−667.089	−0.794509	−0.397255	−	0.917708i	$-0.630037\pi$
$89$	−667.089	−0.794509	−0.397255	+	0.917708i	$0.630037\pi$
$90$	0	0
$91$	−958.407	−1.10405
$92$	−710.996	−0.805723
$93$	0	0
$94$	998.187	1.09527
$95$	327.561	0.353759
$96$	0	0
$97$	−179.654	−0.188052	−0.0940262	−	0.995570i	$-0.529974\pi$
$97$	−179.654	−0.188052	−0.0940262	+	0.995570i	$0.529974\pi$
$98$	−261.679	−0.269730
$99$	0	0
$100$	390.203	0.390203
$101$	−410.736	−0.404651	−0.202325	−	0.979318i	$-0.564850\pi$
$101$	−410.736	−0.404651	−0.202325	+	0.979318i	$0.564850\pi$
$102$	0	0
$103$	1367.66	1.30834	0.654172	−	0.756345i	$-0.273017\pi$
$103$	1367.66	1.30834	0.654172	+	0.756345i	$0.273017\pi$
$104$	−352.228	−0.332104
$105$	0	0
$106$	25.8842	0.0237179
$107$	−395.241	−0.357097	−0.178548	−	0.983931i	$-0.557140\pi$
$107$	−395.241	−0.357097	−0.178548	+	0.983931i	$0.557140\pi$
$108$	0	0
$109$	−505.826	−0.444490	−0.222245	−	0.974991i	$-0.571338\pi$
$109$	−505.826	−0.444490	−0.222245	+	0.974991i	$0.571338\pi$
$110$	0	0
$111$	0	0
$112$	−348.286	−0.293838
$113$	−1537.62	−1.28007	−0.640033	−	0.768347i	$-0.721079\pi$
$113$	−1537.62	−1.28007	−0.640033	+	0.768347i	$0.721079\pi$
$114$	0	0
$115$	2651.68	2.15018
$116$	−597.585	−0.478314
$117$	0	0
$118$	71.1227	0.0554862
$119$	542.335	0.417780
$120$	0	0
$121$	0	0
$122$	1076.69	0.799005
$123$	0	0
$124$	300.574	0.217681
$125$	409.491	0.293008
$126$	0	0
$127$	697.332	0.487230	0.243615	−	0.969872i	$-0.421667\pi$
$127$	697.332	0.487230	0.243615	+	0.969872i	$0.421667\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	1313.65	0.886266
$131$	−259.910	−0.173347	−0.0866735	−	0.996237i	$-0.527624\pi$
$131$	−259.910	−0.173347	−0.0866735	+	0.996237i	$0.527624\pi$
$132$	0	0
$133$	477.962	0.311613
$134$	1039.24	0.669977
$135$	0	0
$136$	199.316	0.125671
$137$	2083.56	1.29935	0.649674	−	0.760213i	$-0.274906\pi$
$137$	2083.56	1.29935	0.649674	+	0.760213i	$0.274906\pi$
$138$	0	0
$139$	−173.429	−0.105828	−0.0529138	−	0.998599i	$-0.516851\pi$
$139$	−173.429	−0.105828	−0.0529138	+	0.998599i	$0.516851\pi$
$140$	1298.94	0.784148
$141$	0	0
$142$	156.897	0.0927220
$143$	0	0
$144$	0	0
$145$	2228.71	1.27645
$146$	2288.14	1.29704
$147$	0	0
$148$	889.346	0.493944
$149$	−449.055	−0.246899	−0.123450	−	0.992351i	$-0.539396\pi$
$149$	−449.055	−0.246899	−0.123450	+	0.992351i	$0.539396\pi$
$150$	0	0
$151$	−28.5358	−0.0153789	−0.00768944	−	0.999970i	$-0.502448\pi$
$151$	−28.5358	−0.0153789	−0.00768944	+	0.999970i	$0.502448\pi$
$152$	175.658	0.0937352
$153$	0	0
$154$	0	0
$155$	−1121.00	−0.580910
$156$	0	0
$157$	1723.86	0.876300	0.438150	−	0.898902i	$-0.355634\pi$
$157$	1723.86	0.876300	0.438150	+	0.898902i	$0.355634\pi$
$158$	1545.09	0.777980
$159$	0	0
$160$	477.380	0.235876
$161$	3869.22	1.89402
$162$	0	0
$163$	3318.22	1.59450	0.797249	−	0.603651i	$-0.206288\pi$
$163$	3318.22	1.59450	0.797249	+	0.603651i	$0.206288\pi$
$164$	−1012.49	−0.482084
$165$	0	0
$166$	−1074.48	−0.502386
$167$	−3015.60	−1.39733	−0.698665	−	0.715449i	$-0.746222\pi$
$167$	−3015.60	−1.39733	−0.698665	+	0.715449i	$0.746222\pi$
$168$	0	0
$169$	−258.486	−0.117654
$170$	−743.356	−0.335369
$171$	0	0
$172$	−522.493	−0.231626
$173$	−2209.41	−0.970973	−0.485486	−	0.874244i	$-0.661357\pi$
$173$	−2209.41	−0.970973	−0.485486	+	0.874244i	$0.661357\pi$
$174$	0	0
$175$	−2123.47	−0.917254
$176$	0	0
$177$	0	0
$178$	1334.18	0.561803
$179$	2271.83	0.948629	0.474315	−	0.880355i	$-0.342696\pi$
$179$	2271.83	0.948629	0.474315	+	0.880355i	$0.342696\pi$
$180$	0	0
$181$	−624.435	−0.256430	−0.128215	−	0.991746i	$-0.540925\pi$
$181$	−624.435	−0.256430	−0.128215	+	0.991746i	$0.540925\pi$
$182$	1916.81	0.780680
$183$	0	0
$184$	1421.99	0.569732
$185$	−3316.85	−1.31816
$186$	0	0
$187$	0	0
$188$	−1996.37	−0.774471
$189$	0	0
$190$	−655.123	−0.250145
$191$	1344.20	0.509229	0.254615	−	0.967043i	$-0.418051\pi$
$191$	1344.20	0.509229	0.254615	+	0.967043i	$0.418051\pi$
$192$	0	0
$193$	−4623.76	−1.72449	−0.862243	−	0.506496i	$-0.830941\pi$
$193$	−4623.76	−1.72449	−0.862243	+	0.506496i	$0.830941\pi$
$194$	359.308	0.132973
$195$	0	0
$196$	523.358	0.190728
$197$	664.691	0.240392	0.120196	−	0.992750i	$-0.461648\pi$
$197$	664.691	0.240392	0.120196	+	0.992750i	$0.461648\pi$
$198$	0	0
$199$	−3042.82	−1.08392	−0.541959	−	0.840405i	$-0.682317\pi$
$199$	−3042.82	−1.08392	−0.541959	+	0.840405i	$0.682317\pi$
$200$	−780.407	−0.275915
$201$	0	0
$202$	821.471	0.286131
$203$	3252.04	1.12438
$204$	0	0
$205$	3776.10	1.28651
$206$	−2735.32	−0.925139
$207$	0	0
$208$	704.457	0.234833
$209$	0	0
$210$	0	0
$211$	−2591.61	−0.845563	−0.422781	−	0.906232i	$-0.638946\pi$
$211$	−2591.61	−0.845563	−0.422781	+	0.906232i	$0.638946\pi$
$212$	−51.7684	−0.0167711
$213$	0	0
$214$	790.481	0.252506
$215$	1948.66	0.618127
$216$	0	0
$217$	−1635.72	−0.511703
$218$	1011.65	0.314302
$219$	0	0
$220$	0	0
$221$	−1096.95	−0.333886
$222$	0	0
$223$	2637.20	0.791929	0.395965	−	0.918266i	$-0.370410\pi$
$223$	2637.20	0.791929	0.395965	+	0.918266i	$0.370410\pi$
$224$	696.571	0.207775
$225$	0	0
$226$	3075.25	0.905143
$227$	250.670	0.0732932	0.0366466	−	0.999328i	$-0.488332\pi$
$227$	250.670	0.0732932	0.0366466	+	0.999328i	$0.488332\pi$
$228$	0	0
$229$	1799.31	0.519222	0.259611	−	0.965713i	$-0.416406\pi$
$229$	1799.31	0.519222	0.259611	+	0.965713i	$0.416406\pi$
$230$	−5303.37	−1.52041
$231$	0	0
$232$	1195.17	0.338219
$233$	3180.48	0.894248	0.447124	−	0.894472i	$-0.352448\pi$
$233$	3180.48	0.894248	0.447124	+	0.894472i	$0.352448\pi$
$234$	0	0
$235$	7445.54	2.06678
$236$	−142.245	−0.0392347
$237$	0	0
$238$	−1084.67	−0.295415
$239$	5013.80	1.35697	0.678485	−	0.734614i	$-0.262637\pi$
$239$	5013.80	1.35697	0.678485	+	0.734614i	$0.262637\pi$
$240$	0	0
$241$	−6074.13	−1.62352	−0.811761	−	0.583990i	$-0.801491\pi$
$241$	−6074.13	−1.62352	−0.811761	+	0.583990i	$0.801491\pi$
$242$	0	0
$243$	0	0
$244$	−2153.37	−0.564982
$245$	−1951.88	−0.508984
$246$	0	0
$247$	−966.746	−0.249039
$248$	−601.149	−0.153923
$249$	0	0
$250$	−818.981	−0.207188
$251$	5569.69	1.40062	0.700310	−	0.713838i	$-0.253045\pi$
$251$	5569.69	1.40062	0.700310	+	0.713838i	$0.253045\pi$
$252$	0	0
$253$	0	0
$254$	−1394.66	−0.344524
$255$	0	0
$256$	256.000	0.0625000
$257$	5907.05	1.43374	0.716871	−	0.697206i	$-0.245574\pi$
$257$	5907.05	1.43374	0.716871	+	0.697206i	$0.245574\pi$
$258$	0	0
$259$	−4839.79	−1.16112
$260$	−2627.30	−0.626685
$261$	0	0
$262$	519.820	0.122575
$263$	−6853.74	−1.60692	−0.803459	−	0.595360i	$-0.797009\pi$
$263$	−6853.74	−1.60692	−0.803459	+	0.595360i	$0.797009\pi$
$264$	0	0
$265$	193.072	0.0447559
$266$	−955.924	−0.220344
$267$	0	0
$268$	−2078.49	−0.473745
$269$	−1101.36	−0.249633	−0.124817	−	0.992180i	$-0.539834\pi$
$269$	−1101.36	−0.249633	−0.124817	+	0.992180i	$0.539834\pi$
$270$	0	0
$271$	2571.69	0.576455	0.288227	−	0.957562i	$-0.406934\pi$
$271$	2571.69	0.576455	0.288227	+	0.957562i	$0.406934\pi$
$272$	−398.632	−0.0888626
$273$	0	0
$274$	−4167.12	−0.918777
$275$	0	0
$276$	0	0
$277$	7340.10	1.59214	0.796072	−	0.605202i	$-0.206908\pi$
$277$	7340.10	1.59214	0.796072	+	0.605202i	$0.206908\pi$
$278$	346.857	0.0748314
$279$	0	0
$280$	−2597.89	−0.554477
$281$	−7492.94	−1.59072	−0.795359	−	0.606139i	$-0.792718\pi$
$281$	−7492.94	−1.59072	−0.795359	+	0.606139i	$0.792718\pi$
$282$	0	0
$283$	6494.10	1.36408	0.682039	−	0.731316i	$-0.261093\pi$
$283$	6494.10	1.36408	0.682039	+	0.731316i	$0.261093\pi$
$284$	−313.795	−0.0655644
$285$	0	0
$286$	0	0
$287$	5509.91	1.13324
$288$	0	0
$289$	−4292.27	−0.873655
$290$	−4457.43	−0.902584
$291$	0	0
$292$	−4576.28	−0.917145
$293$	−2684.87	−0.535331	−0.267666	−	0.963512i	$-0.586252\pi$
$293$	−2684.87	−0.535331	−0.267666	+	0.963512i	$0.586252\pi$
$294$	0	0
$295$	530.509	0.104703
$296$	−1778.69	−0.349271
$297$	0	0
$298$	898.110	0.174584
$299$	−7826.03	−1.51368
$300$	0	0
$301$	2843.39	0.544486
$302$	57.0716	0.0108745
$303$	0	0
$304$	−351.316	−0.0662808
$305$	8031.08	1.50773
$306$	0	0
$307$	−8331.66	−1.54890	−0.774451	−	0.632633i	$-0.781974\pi$
$307$	−8331.66	−1.54890	−0.774451	+	0.632633i	$0.781974\pi$
$308$	0	0
$309$	0	0
$310$	2242.01	0.410766
$311$	5020.35	0.915363	0.457682	−	0.889116i	$-0.348680\pi$
$311$	5020.35	0.915363	0.457682	+	0.889116i	$0.348680\pi$
$312$	0	0
$313$	3022.08	0.545744	0.272872	−	0.962050i	$-0.412026\pi$
$313$	3022.08	0.545744	0.272872	+	0.962050i	$0.412026\pi$
$314$	−3447.72	−0.619637
$315$	0	0
$316$	−3090.18	−0.550115
$317$	10540.1	1.86749	0.933744	−	0.357942i	$-0.116522\pi$
$317$	10540.1	1.86749	0.933744	+	0.357942i	$0.116522\pi$
$318$	0	0
$319$	0	0
$320$	−954.761	−0.166790
$321$	0	0
$322$	−7738.43	−1.33927
$323$	547.054	0.0942381
$324$	0	0
$325$	4295.02	0.733061
$326$	−6636.44	−1.12748
$327$	0	0
$328$	2024.97	0.340885
$329$	10864.2	1.82055
$330$	0	0
$331$	309.871	0.0514563	0.0257281	−	0.999669i	$-0.491810\pi$
$331$	309.871	0.0514563	0.0257281	+	0.999669i	$0.491810\pi$
$332$	2148.97	0.355241
$333$	0	0
$334$	6031.20	0.988061
$335$	7751.78	1.26425
$336$	0	0
$337$	−11164.7	−1.80468	−0.902340	−	0.431024i	$-0.858152\pi$
$337$	−11164.7	−1.80468	−0.902340	+	0.431024i	$0.858152\pi$
$338$	516.972	0.0831941
$339$	0	0
$340$	1486.71	0.237142
$341$	0	0
$342$	0	0
$343$	4618.28	0.727008
$344$	1044.99	0.163785
$345$	0	0
$346$	4418.82	0.686581
$347$	2515.48	0.389159	0.194580	−	0.980887i	$-0.437666\pi$
$347$	2515.48	0.389159	0.194580	+	0.980887i	$0.437666\pi$
$348$	0	0
$349$	11376.6	1.74491	0.872457	−	0.488692i	$-0.162526\pi$
$349$	11376.6	1.74491	0.872457	+	0.488692i	$0.162526\pi$
$350$	4246.94	0.648596
$351$	0	0
$352$	0	0
$353$	−6210.08	−0.936343	−0.468172	−	0.883638i	$-0.655087\pi$
$353$	−6210.08	−0.936343	−0.468172	+	0.883638i	$0.655087\pi$
$354$	0	0
$355$	1170.31	0.174968
$356$	−2668.36	−0.397255
$357$	0	0
$358$	−4543.66	−0.670782
$359$	−2487.51	−0.365698	−0.182849	−	0.983141i	$-0.558532\pi$
$359$	−2487.51	−0.365698	−0.182849	+	0.983141i	$0.558532\pi$
$360$	0	0
$361$	−6376.88	−0.929710
$362$	1248.87	0.181324
$363$	0	0
$364$	−3833.63	−0.552024
$365$	17067.4	2.44753
$366$	0	0
$367$	6719.53	0.955739	0.477870	−	0.878431i	$-0.341409\pi$
$367$	6719.53	0.955739	0.477870	+	0.878431i	$0.341409\pi$
$368$	−2843.98	−0.402861
$369$	0	0
$370$	6633.69	0.932079
$371$	281.722	0.0394239
$372$	0	0
$373$	8614.37	1.19580	0.597902	−	0.801569i	$-0.296001\pi$
$373$	8614.37	1.19580	0.597902	+	0.801569i	$0.296001\pi$
$374$	0	0
$375$	0	0
$376$	3992.75	0.547634
$377$	−6577.70	−0.898592
$378$	0	0
$379$	−8009.10	−1.08549	−0.542744	−	0.839898i	$-0.682614\pi$
$379$	−8009.10	−1.08549	−0.542744	+	0.839898i	$0.682614\pi$
$380$	1310.25	0.176879
$381$	0	0
$382$	−2688.40	−0.360079
$383$	−8011.95	−1.06891	−0.534454	−	0.845198i	$-0.679483\pi$
$383$	−8011.95	−1.06891	−0.534454	+	0.845198i	$0.679483\pi$
$384$	0	0
$385$	0	0
$386$	9247.52	1.21940
$387$	0	0
$388$	−718.615	−0.0940262
$389$	−8029.60	−1.04657	−0.523287	−	0.852157i	$-0.675294\pi$
$389$	−8029.60	−1.04657	−0.523287	+	0.852157i	$0.675294\pi$
$390$	0	0
$391$	4428.53	0.572789
$392$	−1046.72	−0.134865
$393$	0	0
$394$	−1329.38	−0.169983
$395$	11524.9	1.46806
$396$	0	0
$397$	4435.00	0.560670	0.280335	−	0.959902i	$-0.409554\pi$
$397$	4435.00	0.560670	0.280335	+	0.959902i	$0.409554\pi$
$398$	6085.64	0.766446
$399$	0	0
$400$	1560.81	0.195102
$401$	3929.21	0.489315	0.244657	−	0.969610i	$-0.421325\pi$
$401$	3929.21	0.489315	0.244657	+	0.969610i	$0.421325\pi$
$402$	0	0
$403$	3308.46	0.408949
$404$	−1642.94	−0.202325
$405$	0	0
$406$	−6504.07	−0.795054
$407$	0	0
$408$	0	0
$409$	14564.2	1.76077	0.880384	−	0.474261i	$-0.157285\pi$
$409$	14564.2	1.76077	0.880384	+	0.474261i	$0.157285\pi$
$410$	−7552.20	−0.909698
$411$	0	0
$412$	5470.64	0.654172
$413$	774.094	0.0922293
$414$	0	0
$415$	−8014.65	−0.948008
$416$	−1408.91	−0.166052
$417$	0	0
$418$	0	0
$419$	−4028.77	−0.469734	−0.234867	−	0.972028i	$-0.575465\pi$
$419$	−4028.77	−0.469734	−0.234867	+	0.972028i	$0.575465\pi$
$420$	0	0
$421$	−8527.60	−0.987197	−0.493599	−	0.869690i	$-0.664319\pi$
$421$	−8527.60	−0.987197	−0.493599	+	0.869690i	$0.664319\pi$
$422$	5183.22	0.597903
$423$	0	0
$424$	103.537	0.0118589
$425$	−2430.43	−0.277396
$426$	0	0
$427$	11718.6	1.32811
$428$	−1580.96	−0.178548
$429$	0	0
$430$	−3897.31	−0.437082
$431$	13415.2	1.49928	0.749639	−	0.661847i	$-0.230227\pi$
$431$	13415.2	1.49928	0.749639	+	0.661847i	$0.230227\pi$
$432$	0	0
$433$	4132.31	0.458628	0.229314	−	0.973352i	$-0.426352\pi$
$433$	4132.31	0.458628	0.229314	+	0.973352i	$0.426352\pi$
$434$	3271.43	0.361829
$435$	0	0
$436$	−2023.30	−0.222245
$437$	3902.88	0.427231
$438$	0	0
$439$	−3358.46	−0.365126	−0.182563	−	0.983194i	$-0.558439\pi$
$439$	−3358.46	−0.365126	−0.182563	+	0.983194i	$0.558439\pi$
$440$	0	0
$441$	0	0
$442$	2193.90	0.236093
$443$	441.749	0.0473773	0.0236886	−	0.999719i	$-0.492459\pi$
$443$	441.749	0.0473773	0.0236886	+	0.999719i	$0.492459\pi$
$444$	0	0
$445$	9951.73	1.06013
$446$	−5274.41	−0.559979
$447$	0	0
$448$	−1393.14	−0.146919
$449$	408.478	0.0429338	0.0214669	−	0.999770i	$-0.493166\pi$
$449$	408.478	0.0429338	0.0214669	+	0.999770i	$0.493166\pi$
$450$	0	0
$451$	0	0
$452$	−6150.49	−0.640033
$453$	0	0
$454$	−501.340	−0.0518261
$455$	14297.6	1.47315
$456$	0	0
$457$	−1522.85	−0.155877	−0.0779384	−	0.996958i	$-0.524834\pi$
$457$	−1522.85	−0.155877	−0.0779384	+	0.996958i	$0.524834\pi$
$458$	−3598.63	−0.367146
$459$	0	0
$460$	10606.7	1.07509
$461$	−13861.4	−1.40041	−0.700203	−	0.713944i	$-0.746907\pi$
$461$	−13861.4	−1.40041	−0.700203	+	0.713944i	$0.746907\pi$
$462$	0	0
$463$	−6502.26	−0.652669	−0.326334	−	0.945254i	$-0.605814\pi$
$463$	−6502.26	−0.652669	−0.326334	+	0.945254i	$0.605814\pi$
$464$	−2390.34	−0.239157
$465$	0	0
$466$	−6360.95	−0.632329
$467$	−423.973	−0.0420110	−0.0210055	−	0.999779i	$-0.506687\pi$
$467$	−423.973	−0.0420110	−0.0210055	+	0.999779i	$0.506687\pi$
$468$	0	0
$469$	11311.0	1.11364
$470$	−14891.1	−1.46144
$471$	0	0
$472$	284.491	0.0277431
$473$	0	0
$474$	0	0
$475$	−2141.95	−0.206904
$476$	2169.34	0.208890
$477$	0	0
$478$	−10027.6	−0.959523
$479$	4557.34	0.434718	0.217359	−	0.976092i	$-0.430256\pi$
$479$	4557.34	0.434718	0.217359	+	0.976092i	$0.430256\pi$
$480$	0	0
$481$	9789.15	0.927956
$482$	12148.3	1.14800
$483$	0	0
$484$	0	0
$485$	2680.10	0.250922
$486$	0	0
$487$	−6708.92	−0.624251	−0.312126	−	0.950041i	$-0.601041\pi$
$487$	−6708.92	−0.624251	−0.312126	+	0.950041i	$0.601041\pi$
$488$	4306.75	0.399503
$489$	0	0
$490$	3903.76	0.359906
$491$	14527.7	1.33528	0.667641	−	0.744483i	$-0.267304\pi$
$491$	14527.7	1.33528	0.667641	+	0.744483i	$0.267304\pi$
$492$	0	0
$493$	3722.13	0.340034
$494$	1933.49	0.176097
$495$	0	0
$496$	1202.30	0.108840
$497$	1707.66	0.154123
$498$	0	0
$499$	−9699.53	−0.870162	−0.435081	−	0.900391i	$-0.643280\pi$
$499$	−9699.53	−0.870162	−0.435081	+	0.900391i	$0.643280\pi$
$500$	1637.96	0.146504
$501$	0	0
$502$	−11139.4	−0.990388
$503$	15707.1	1.39234	0.696170	−	0.717877i	$-0.254886\pi$
$503$	15707.1	1.39234	0.696170	+	0.717877i	$0.254886\pi$
$504$	0	0
$505$	6127.41	0.539933
$506$	0	0
$507$	0	0
$508$	2789.33	0.243615
$509$	1474.08	0.128364	0.0641822	−	0.997938i	$-0.479556\pi$
$509$	1474.08	0.128364	0.0641822	+	0.997938i	$0.479556\pi$
$510$	0	0
$511$	24903.9	2.15594
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−11814.1	−1.01381
$515$	−20402.9	−1.74575
$516$	0	0
$517$	0	0
$518$	9679.57	0.821035
$519$	0	0
$520$	5254.59	0.443133
$521$	9272.84	0.779752	0.389876	−	0.920867i	$-0.372518\pi$
$521$	9272.84	0.779752	0.389876	+	0.920867i	$0.372518\pi$
$522$	0	0
$523$	4242.50	0.354707	0.177353	−	0.984147i	$-0.443246\pi$
$523$	4242.50	0.354707	0.177353	+	0.984147i	$0.443246\pi$
$524$	−1039.64	−0.0866735
$525$	0	0
$526$	13707.5	1.13626
$527$	−1872.17	−0.154749
$528$	0	0
$529$	19427.7	1.59676
$530$	−386.144	−0.0316472
$531$	0	0
$532$	1911.85	0.155807
$533$	−11144.6	−0.905675
$534$	0	0
$535$	5896.25	0.476481
$536$	4156.97	0.334988
$537$	0	0
$538$	2202.73	0.176517
$539$	0	0
$540$	0	0
$541$	6520.25	0.518166	0.259083	−	0.965855i	$-0.416580\pi$
$541$	6520.25	0.518166	0.259083	+	0.965855i	$0.416580\pi$
$542$	−5143.39	−0.407615
$543$	0	0
$544$	797.264	0.0628353
$545$	7545.98	0.593091
$546$	0	0
$547$	−6370.78	−0.497980	−0.248990	−	0.968506i	$-0.580099\pi$
$547$	−6370.78	−0.497980	−0.248990	+	0.968506i	$0.580099\pi$
$548$	8334.24	0.649674
$549$	0	0
$550$	0	0
$551$	3280.33	0.253624
$552$	0	0
$553$	16816.7	1.29316
$554$	−14680.2	−1.12582
$555$	0	0
$556$	−693.715	−0.0529138
$557$	6103.99	0.464335	0.232167	−	0.972676i	$-0.425418\pi$
$557$	6103.99	0.464335	0.232167	+	0.972676i	$0.425418\pi$
$558$	0	0
$559$	−5751.15	−0.435148
$560$	5195.77	0.392074
$561$	0	0
$562$	14985.9	1.12481
$563$	−13592.6	−1.01751	−0.508756	−	0.860911i	$-0.669894\pi$
$563$	−13592.6	−1.01751	−0.508756	+	0.860911i	$0.669894\pi$
$564$	0	0
$565$	22938.5	1.70802
$566$	−12988.2	−0.964549
$567$	0	0
$568$	627.589	0.0463610
$569$	17197.6	1.26706	0.633532	−	0.773717i	$-0.281605\pi$
$569$	17197.6	1.26706	0.633532	+	0.773717i	$0.281605\pi$
$570$	0	0
$571$	−2475.65	−0.181441	−0.0907203	−	0.995876i	$-0.528917\pi$
$571$	−2475.65	−0.181441	−0.0907203	+	0.995876i	$0.528917\pi$
$572$	0	0
$573$	0	0
$574$	−11019.8	−0.801321
$575$	−17339.6	−1.25758
$576$	0	0
$577$	−20339.7	−1.46751	−0.733755	−	0.679414i	$-0.762234\pi$
$577$	−20339.7	−1.46751	−0.733755	+	0.679414i	$0.762234\pi$
$578$	8584.54	0.617767
$579$	0	0
$580$	8914.86	0.638223
$581$	−11694.6	−0.835067
$582$	0	0
$583$	0	0
$584$	9152.55	0.648519
$585$	0	0
$586$	5369.75	0.378536
$587$	14818.4	1.04195	0.520973	−	0.853573i	$-0.325569\pi$
$587$	14818.4	1.04195	0.520973	+	0.853573i	$0.325569\pi$
$588$	0	0
$589$	−1649.95	−0.115424
$590$	−1061.02	−0.0740363
$591$	0	0
$592$	3557.38	0.246972
$593$	11123.2	0.770281	0.385140	−	0.922858i	$-0.374153\pi$
$593$	11123.2	0.770281	0.385140	+	0.922858i	$0.374153\pi$
$594$	0	0
$595$	−8090.63	−0.557451
$596$	−1796.22	−0.123450
$597$	0	0
$598$	15652.1	1.07034
$599$	−8285.18	−0.565147	−0.282574	−	0.959246i	$-0.591188\pi$
$599$	−8285.18	−0.565147	−0.282574	+	0.959246i	$0.591188\pi$
$600$	0	0
$601$	−27318.3	−1.85414	−0.927069	−	0.374890i	$-0.877680\pi$
$601$	−27318.3	−1.85414	−0.927069	+	0.374890i	$0.877680\pi$
$602$	−5686.78	−0.385009
$603$	0	0
$604$	−114.143	−0.00768944
$605$	0	0
$606$	0	0
$607$	−16327.7	−1.09180	−0.545898	−	0.837851i	$-0.683812\pi$
$607$	−16327.7	−1.09180	−0.545898	+	0.837851i	$0.683812\pi$
$608$	702.632	0.0468676
$609$	0	0
$610$	−16062.2	−1.06613
$611$	−21974.4	−1.45497
$612$	0	0
$613$	21025.3	1.38533	0.692663	−	0.721262i	$-0.256437\pi$
$613$	21025.3	1.38533	0.692663	+	0.721262i	$0.256437\pi$
$614$	16663.3	1.09524
$615$	0	0
$616$	0	0
$617$	−871.824	−0.0568854	−0.0284427	−	0.999595i	$-0.509055\pi$
$617$	−871.824	−0.0568854	−0.0284427	+	0.999595i	$0.509055\pi$
$618$	0	0
$619$	9397.52	0.610207	0.305104	−	0.952319i	$-0.401309\pi$
$619$	9397.52	0.610207	0.305104	+	0.952319i	$0.401309\pi$
$620$	−4484.01	−0.290455
$621$	0	0
$622$	−10040.7	−0.647260
$623$	14521.1	0.933829
$624$	0	0
$625$	−18302.7	−1.17137
$626$	−6044.15	−0.385899
$627$	0	0
$628$	6895.44	0.438150
$629$	−5539.40	−0.351145
$630$	0	0
$631$	−14535.9	−0.917062	−0.458531	−	0.888678i	$-0.651624\pi$
$631$	−14535.9	−0.917062	−0.458531	+	0.888678i	$0.651624\pi$
$632$	6180.37	0.388990
$633$	0	0
$634$	−21080.3	−1.32051
$635$	−10402.9	−0.650120
$636$	0	0
$637$	5760.67	0.358314
$638$	0	0
$639$	0	0
$640$	1909.52	0.117938
$641$	11419.7	0.703666	0.351833	−	0.936063i	$-0.385559\pi$
$641$	11419.7	0.703666	0.351833	+	0.936063i	$0.385559\pi$
$642$	0	0
$643$	23969.9	1.47011	0.735055	−	0.678007i	$-0.237156\pi$
$643$	23969.9	1.47011	0.735055	+	0.678007i	$0.237156\pi$
$644$	15476.9	0.947009
$645$	0	0
$646$	−1094.11	−0.0666364
$647$	−8314.25	−0.505204	−0.252602	−	0.967570i	$-0.581286\pi$
$647$	−8314.25	−0.505204	−0.252602	+	0.967570i	$0.581286\pi$
$648$	0	0
$649$	0	0
$650$	−8590.04	−0.518353
$651$	0	0
$652$	13272.9	0.797249
$653$	30079.3	1.80260	0.901298	−	0.433200i	$-0.142616\pi$
$653$	30079.3	1.80260	0.901298	+	0.433200i	$0.142616\pi$
$654$	0	0
$655$	3877.38	0.231300
$656$	−4049.94	−0.241042
$657$	0	0
$658$	−21728.4	−1.28733
$659$	−10041.6	−0.593572	−0.296786	−	0.954944i	$-0.595915\pi$
$659$	−10041.6	−0.593572	−0.296786	+	0.954944i	$0.595915\pi$
$660$	0	0
$661$	1402.50	0.0825281	0.0412640	−	0.999148i	$-0.486862\pi$
$661$	1402.50	0.0825281	0.0412640	+	0.999148i	$0.486862\pi$
$662$	−619.741	−0.0363851
$663$	0	0
$664$	−4297.93	−0.251193
$665$	−7130.31	−0.415792
$666$	0	0
$667$	26555.1	1.54155
$668$	−12062.4	−0.698665
$669$	0	0
$670$	−15503.6	−0.893963
$671$	0	0
$672$	0	0
$673$	−20385.4	−1.16761	−0.583803	−	0.811895i	$-0.698436\pi$
$673$	−20385.4	−1.16761	−0.583803	+	0.811895i	$0.698436\pi$
$674$	22329.3	1.27610
$675$	0	0
$676$	−1033.94	−0.0588271
$677$	−7385.00	−0.419245	−0.209622	−	0.977782i	$-0.567223\pi$
$677$	−7385.00	−0.419245	−0.209622	+	0.977782i	$0.567223\pi$
$678$	0	0
$679$	3910.68	0.221028
$680$	−2973.42	−0.167685
$681$	0	0
$682$	0	0
$683$	25844.0	1.44787	0.723935	−	0.689868i	$-0.242332\pi$
$683$	25844.0	1.44787	0.723935	+	0.689868i	$0.242332\pi$
$684$	0	0
$685$	−31082.8	−1.73374
$686$	−9236.56	−0.514072
$687$	0	0
$688$	−2089.97	−0.115813
$689$	−569.822	−0.0315073
$690$	0	0
$691$	8438.80	0.464583	0.232292	−	0.972646i	$-0.425378\pi$
$691$	8438.80	0.464583	0.232292	+	0.972646i	$0.425378\pi$
$692$	−8837.64	−0.485486
$693$	0	0
$694$	−5030.97	−0.275177
$695$	2587.23	0.141208
$696$	0	0
$697$	6306.39	0.342714
$698$	−22753.2	−1.23384
$699$	0	0
$700$	−8493.89	−0.458627
$701$	−12983.4	−0.699536	−0.349768	−	0.936836i	$-0.613740\pi$
$701$	−12983.4	−0.699536	−0.349768	+	0.936836i	$0.613740\pi$
$702$	0	0
$703$	−4881.90	−0.261912
$704$	0	0
$705$	0	0
$706$	12420.2	0.662095
$707$	8940.83	0.475608
$708$	0	0
$709$	−17918.6	−0.949150	−0.474575	−	0.880215i	$-0.657398\pi$
$709$	−17918.6	−0.949150	−0.474575	+	0.880215i	$0.657398\pi$
$710$	−2340.61	−0.123721
$711$	0	0
$712$	5336.71	0.280901
$713$	−13356.7	−0.701560
$714$	0	0
$715$	0	0
$716$	9087.33	0.474315
$717$	0	0
$718$	4975.01	0.258587
$719$	−3517.35	−0.182441	−0.0912204	−	0.995831i	$-0.529077\pi$
$719$	−3517.35	−0.182441	−0.0912204	+	0.995831i	$0.529077\pi$
$720$	0	0
$721$	−29771.0	−1.53777
$722$	12753.8	0.657404
$723$	0	0
$724$	−2497.74	−0.128215
$725$	−14573.7	−0.746559
$726$	0	0
$727$	−29438.9	−1.50183	−0.750913	−	0.660401i	$-0.770386\pi$
$727$	−29438.9	−1.50183	−0.750913	+	0.660401i	$0.770386\pi$
$728$	7667.26	0.390340
$729$	0	0
$730$	−34134.8	−1.73066
$731$	3254.41	0.164663
$732$	0	0
$733$	−3435.90	−0.173135	−0.0865673	−	0.996246i	$-0.527590\pi$
$733$	−3435.90	−0.173135	−0.0865673	+	0.996246i	$0.527590\pi$
$734$	−13439.1	−0.675810
$735$	0	0
$736$	5687.97	0.284866
$737$	0	0
$738$	0	0
$739$	33601.8	1.67261	0.836307	−	0.548262i	$-0.184710\pi$
$739$	33601.8	1.67261	0.836307	+	0.548262i	$0.184710\pi$
$740$	−13267.4	−0.659079
$741$	0	0
$742$	−563.444	−0.0278769
$743$	−1193.70	−0.0589404	−0.0294702	−	0.999566i	$-0.509382\pi$
$743$	−1193.70	−0.0589404	−0.0294702	+	0.999566i	$0.509382\pi$
$744$	0	0
$745$	6699.06	0.329443
$746$	−17228.7	−0.845562
$747$	0	0
$748$	0	0
$749$	8603.54	0.419715
$750$	0	0
$751$	24509.4	1.19089	0.595446	−	0.803396i	$-0.296976\pi$
$751$	24509.4	1.19089	0.595446	+	0.803396i	$0.296976\pi$
$752$	−7985.50	−0.387235
$753$	0	0
$754$	13155.4	0.635400
$755$	425.701	0.0205203
$756$	0	0
$757$	10803.8	0.518721	0.259360	−	0.965781i	$-0.416488\pi$
$757$	10803.8	0.518721	0.259360	+	0.965781i	$0.416488\pi$
$758$	16018.2	0.767555
$759$	0	0
$760$	−2620.49	−0.125073
$761$	8175.91	0.389457	0.194728	−	0.980857i	$-0.437617\pi$
$761$	8175.91	0.389457	0.194728	+	0.980857i	$0.437617\pi$
$762$	0	0
$763$	11010.8	0.522432
$764$	5376.79	0.254615
$765$	0	0
$766$	16023.9	0.755831
$767$	−1565.72	−0.0737089
$768$	0	0
$769$	28895.9	1.35502	0.677511	−	0.735513i	$-0.263059\pi$
$769$	28895.9	1.35502	0.677511	+	0.735513i	$0.263059\pi$
$770$	0	0
$771$	0	0
$772$	−18495.0	−0.862243
$773$	17996.8	0.837387	0.418693	−	0.908128i	$-0.362488\pi$
$773$	17996.8	0.837387	0.418693	+	0.908128i	$0.362488\pi$
$774$	0	0
$775$	7330.32	0.339759
$776$	1437.23	0.0664866
$777$	0	0
$778$	16059.2	0.740039
$779$	5557.85	0.255623
$780$	0	0
$781$	0	0
$782$	−8857.06	−0.405023
$783$	0	0
$784$	2093.43	0.0953640
$785$	−25716.8	−1.16926
$786$	0	0
$787$	15876.7	0.719114	0.359557	−	0.933123i	$-0.382928\pi$
$787$	15876.7	0.719114	0.359557	+	0.933123i	$0.382928\pi$
$788$	2658.76	0.120196
$789$	0	0
$790$	−23049.9	−1.03807
$791$	33470.8	1.50453
$792$	0	0
$793$	−23702.5	−1.06141
$794$	−8869.99	−0.396454
$795$	0	0
$796$	−12171.3	−0.541959
$797$	−16497.7	−0.733223	−0.366611	−	0.930374i	$-0.619482\pi$
$797$	−16497.7	−0.733223	−0.366611	+	0.930374i	$0.619482\pi$
$798$	0	0
$799$	12434.7	0.550572
$800$	−3121.63	−0.137958
$801$	0	0
$802$	−7858.41	−0.345998
$803$	0	0
$804$	0	0
$805$	−57721.5	−2.52722
$806$	−6616.93	−0.289170
$807$	0	0
$808$	3285.88	0.143066
$809$	−9102.95	−0.395603	−0.197801	−	0.980242i	$-0.563380\pi$
$809$	−9102.95	−0.395603	−0.197801	+	0.980242i	$0.563380\pi$
$810$	0	0
$811$	−36576.2	−1.58368	−0.791841	−	0.610728i	$-0.790877\pi$
$811$	−36576.2	−1.58368	−0.791841	+	0.610728i	$0.790877\pi$
$812$	13008.1	0.562188
$813$	0	0
$814$	0	0
$815$	−49501.7	−2.12757
$816$	0	0
$817$	2868.13	0.122819
$818$	−29128.4	−1.24505
$819$	0	0
$820$	15104.4	0.643254
$821$	−3644.21	−0.154913	−0.0774567	−	0.996996i	$-0.524680\pi$
$821$	−3644.21	−0.154913	−0.0774567	+	0.996996i	$0.524680\pi$
$822$	0	0
$823$	−16895.9	−0.715618	−0.357809	−	0.933795i	$-0.616476\pi$
$823$	−16895.9	−0.715618	−0.357809	+	0.933795i	$0.616476\pi$
$824$	−10941.3	−0.462570
$825$	0	0
$826$	−1548.19	−0.0652160
$827$	−38778.1	−1.63053	−0.815263	−	0.579090i	$-0.803408\pi$
$827$	−38778.1	−1.63053	−0.815263	+	0.579090i	$0.803408\pi$
$828$	0	0
$829$	−12394.3	−0.519267	−0.259634	−	0.965707i	$-0.583602\pi$
$829$	−12394.3	−0.519267	−0.259634	+	0.965707i	$0.583602\pi$
$830$	16029.3	0.670343
$831$	0	0
$832$	2817.83	0.117417
$833$	−3259.80	−0.135589
$834$	0	0
$835$	44987.1	1.86448
$836$	0	0
$837$	0	0
$838$	8057.55	0.332152
$839$	−24563.8	−1.01077	−0.505386	−	0.862893i	$-0.668650\pi$
$839$	−24563.8	−1.01077	−0.505386	+	0.862893i	$0.668650\pi$
$840$	0	0
$841$	−2069.74	−0.0848635
$842$	17055.2	0.698054
$843$	0	0
$844$	−10366.4	−0.422781
$845$	3856.13	0.156988
$846$	0	0
$847$	0	0
$848$	−207.074	−0.00838554
$849$	0	0
$850$	4860.86	0.196148
$851$	−39520.1	−1.59193
$852$	0	0
$853$	12207.6	0.490013	0.245007	−	0.969521i	$-0.421210\pi$
$853$	12207.6	0.490013	0.245007	+	0.969521i	$0.421210\pi$
$854$	−23437.2	−0.939114
$855$	0	0
$856$	3161.93	0.126253
$857$	7281.72	0.290244	0.145122	−	0.989414i	$-0.453643\pi$
$857$	7281.72	0.290244	0.145122	+	0.989414i	$0.453643\pi$
$858$	0	0
$859$	5927.39	0.235437	0.117718	−	0.993047i	$-0.462442\pi$
$859$	5927.39	0.235437	0.117718	+	0.993047i	$0.462442\pi$
$860$	7794.62	0.309063
$861$	0	0
$862$	−26830.4	−1.06015
$863$	38561.9	1.52104	0.760522	−	0.649312i	$-0.224943\pi$
$863$	38561.9	1.52104	0.760522	+	0.649312i	$0.224943\pi$
$864$	0	0
$865$	32960.3	1.29559
$866$	−8264.62	−0.324299
$867$	0	0
$868$	−6542.86	−0.255852
$869$	0	0
$870$	0	0
$871$	−22878.2	−0.890009
$872$	4046.61	0.157151
$873$	0	0
$874$	−7805.76	−0.302098
$875$	−8913.73	−0.344388
$876$	0	0
$877$	−28512.7	−1.09784	−0.548919	−	0.835875i	$-0.684961\pi$
$877$	−28512.7	−1.09784	−0.548919	+	0.835875i	$0.684961\pi$
$878$	6716.91	0.258183
$879$	0	0
$880$	0	0
$881$	−40747.6	−1.55826	−0.779128	−	0.626865i	$-0.784338\pi$
$881$	−40747.6	−1.55826	−0.779128	+	0.626865i	$0.784338\pi$
$882$	0	0
$883$	−3595.59	−0.137034	−0.0685171	−	0.997650i	$-0.521827\pi$
$883$	−3595.59	−0.137034	−0.0685171	+	0.997650i	$0.521827\pi$
$884$	−4387.80	−0.166943
$885$	0	0
$886$	−883.499	−0.0335008
$887$	−7715.30	−0.292057	−0.146028	−	0.989280i	$-0.546649\pi$
$887$	−7715.30	−0.292057	−0.146028	+	0.989280i	$0.546649\pi$
$888$	0	0
$889$	−15179.4	−0.572667
$890$	−19903.5	−0.749624
$891$	0	0
$892$	10548.8	0.395965
$893$	10958.7	0.410660
$894$	0	0
$895$	−33891.5	−1.26577
$896$	2786.29	0.103888
$897$	0	0
$898$	−816.957	−0.0303588
$899$	−11226.2	−0.416478
$900$	0	0
$901$	322.446	0.0119226
$902$	0	0
$903$	0	0
$904$	12301.0	0.452572
$905$	9315.41	0.342160
$906$	0	0
$907$	−33713.7	−1.23423	−0.617114	−	0.786874i	$-0.711698\pi$
$907$	−33713.7	−1.23423	−0.617114	+	0.786874i	$0.711698\pi$
$908$	1002.68	0.0366466
$909$	0	0
$910$	−28595.3	−1.04168
$911$	−15030.7	−0.546642	−0.273321	−	0.961923i	$-0.588122\pi$
$911$	−15030.7	−0.546642	−0.273321	+	0.961923i	$0.588122\pi$
$912$	0	0
$913$	0	0
$914$	3045.69	0.110222
$915$	0	0
$916$	7197.25	0.259611
$917$	5657.69	0.203744
$918$	0	0
$919$	17090.7	0.613460	0.306730	−	0.951797i	$-0.400765\pi$
$919$	17090.7	0.613460	0.306730	+	0.951797i	$0.400765\pi$
$920$	−21213.5	−0.760204
$921$	0	0
$922$	27722.7	0.990237
$923$	−3453.98	−0.123173
$924$	0	0
$925$	21689.1	0.770955
$926$	13004.5	0.461506
$927$	0	0
$928$	4780.68	0.169109
$929$	−13160.3	−0.464773	−0.232386	−	0.972624i	$-0.574653\pi$
$929$	−13160.3	−0.464773	−0.232386	+	0.972624i	$0.574653\pi$
$930$	0	0
$931$	−2872.87	−0.101133
$932$	12721.9	0.447124
$933$	0	0
$934$	847.946	0.0297063
$935$	0	0
$936$	0	0
$937$	−4056.29	−0.141423	−0.0707114	−	0.997497i	$-0.522527\pi$
$937$	−4056.29	−0.141423	−0.0707114	+	0.997497i	$0.522527\pi$
$938$	−22622.1	−0.787460
$939$	0	0
$940$	29782.2	1.03339
$941$	33397.6	1.15699	0.578497	−	0.815684i	$-0.303639\pi$
$941$	33397.6	1.15699	0.578497	+	0.815684i	$0.303639\pi$
$942$	0	0
$943$	44992.1	1.55370
$944$	−568.982	−0.0196173
$945$	0	0
$946$	0	0
$947$	−25660.8	−0.880533	−0.440267	−	0.897867i	$-0.645116\pi$
$947$	−25660.8	−0.880533	−0.440267	+	0.897867i	$0.645116\pi$
$948$	0	0
$949$	−50371.7	−1.72301
$950$	4283.90	0.146303
$951$	0	0
$952$	−4338.68	−0.147707
$953$	48852.8	1.66054	0.830271	−	0.557359i	$-0.188185\pi$
$953$	48852.8	1.66054	0.830271	+	0.557359i	$0.188185\pi$
$954$	0	0
$955$	−20052.9	−0.679474
$956$	20055.2	0.678485
$957$	0	0
$958$	−9114.68	−0.307392
$959$	−45354.6	−1.52719
$960$	0	0
$961$	−24144.4	−0.810461
$962$	−19578.3	−0.656164
$963$	0	0
$964$	−24296.5	−0.811761
$965$	68977.9	2.30101
$966$	0	0
$967$	48861.8	1.62491	0.812456	−	0.583023i	$-0.198130\pi$
$967$	48861.8	1.62491	0.812456	+	0.583023i	$0.198130\pi$
$968$	0	0
$969$	0	0
$970$	−5360.20	−0.177429
$971$	−34887.5	−1.15303	−0.576516	−	0.817086i	$-0.695588\pi$
$971$	−34887.5	−1.15303	−0.576516	+	0.817086i	$0.695588\pi$
$972$	0	0
$973$	3775.17	0.124385
$974$	13417.8	0.441412
$975$	0	0
$976$	−8613.49	−0.282491
$977$	19012.4	0.622580	0.311290	−	0.950315i	$-0.399239\pi$
$977$	19012.4	0.622580	0.311290	+	0.950315i	$0.399239\pi$
$978$	0	0
$979$	0	0
$980$	−7807.52	−0.254492
$981$	0	0
$982$	−29055.3	−0.944187
$983$	7447.89	0.241659	0.120829	−	0.992673i	$-0.461445\pi$
$983$	7447.89	0.241659	0.120829	+	0.992673i	$0.461445\pi$
$984$	0	0
$985$	−9915.95	−0.320760
$986$	−7444.27	−0.240440
$987$	0	0
$988$	−3866.98	−0.124519
$989$	23218.2	0.746506
$990$	0	0
$991$	5313.37	0.170318	0.0851588	−	0.996367i	$-0.472860\pi$
$991$	5313.37	0.170318	0.0851588	+	0.996367i	$0.472860\pi$
$992$	−2404.60	−0.0769617
$993$	0	0
$994$	−3415.32	−0.108981
$995$	45393.2	1.44629
$996$	0	0
$997$	−53523.9	−1.70022	−0.850110	−	0.526604i	$-0.823465\pi$
$997$	−53523.9	−1.70022	−0.850110	+	0.526604i	$0.823465\pi$
$998$	19399.1	0.615297
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.bt.1.1		4
3.2	odd	2		242.4.a.o.1.1		4
11.7	odd	10		198.4.f.d.181.1		8
11.8	odd	10		198.4.f.d.163.1		8
11.10	odd	2		2178.4.a.by.1.1		4
12.11	even	2		1936.4.a.bm.1.4		4
33.2	even	10		242.4.c.r.81.1		8
33.5	odd	10		242.4.c.n.3.1		8
33.8	even	10		22.4.c.b.9.2	yes	8
33.14	odd	10		242.4.c.q.9.2		8
33.17	even	10		242.4.c.r.3.1		8
33.20	odd	10		242.4.c.n.81.1		8
33.26	odd	10		242.4.c.q.27.2		8
33.29	even	10		22.4.c.b.5.2	✓	8
33.32	even	2		242.4.a.n.1.1		4
132.95	odd	10		176.4.m.b.49.1		8
132.107	odd	10		176.4.m.b.97.1		8
132.131	odd	2		1936.4.a.bn.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.5.2	✓	8	33.29	even	10
22.4.c.b.9.2	yes	8	33.8	even	10
176.4.m.b.49.1		8	132.95	odd	10
176.4.m.b.97.1		8	132.107	odd	10
198.4.f.d.163.1		8	11.8	odd	10
198.4.f.d.181.1		8	11.7	odd	10
242.4.a.n.1.1		4	33.32	even	2
242.4.a.o.1.1		4	3.2	odd	2
242.4.c.n.3.1		8	33.5	odd	10
242.4.c.n.81.1		8	33.20	odd	10
242.4.c.q.9.2		8	33.14	odd	10
242.4.c.q.27.2		8	33.26	odd	10
242.4.c.r.3.1		8	33.17	even	10
242.4.c.r.81.1		8	33.2	even	10
1936.4.a.bm.1.4		4	12.11	even	2
1936.4.a.bn.1.4		4	132.131	odd	2
2178.4.a.bt.1.1		4	1.1	even	1	trivial
2178.4.a.by.1.1		4	11.10	odd	2

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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -14.9181 q^{5} -21.7679 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -14.9181 q^{5} -21.7679 q^{7} -8.00000 q^{8} +29.8363 q^{10} +44.0286 q^{13} +43.5357 q^{14} +16.0000 q^{16} -24.9145 q^{17} -21.9573 q^{19} -59.6725 q^{20} -177.749 q^{23} +97.5508 q^{25} -88.0571 q^{26} -87.0714 q^{28} -149.396 q^{29} +75.1436 q^{31} -32.0000 q^{32} +49.8290 q^{34} +324.736 q^{35} +222.336 q^{37} +43.9145 q^{38} +119.345 q^{40} -253.121 q^{41} -130.623 q^{43} +355.498 q^{46} -499.093 q^{47} +130.839 q^{49} -195.102 q^{50} +176.114 q^{52} -12.9421 q^{53} +174.143 q^{56} +298.793 q^{58} -35.5614 q^{59} -538.343 q^{61} -150.287 q^{62} +64.0000 q^{64} -656.824 q^{65} -519.621 q^{67} -99.6580 q^{68} -649.472 q^{70} -78.4486 q^{71} -1144.07 q^{73} -444.673 q^{74} -87.8290 q^{76} -772.546 q^{79} -238.690 q^{80} +506.243 q^{82} +537.242 q^{83} +371.678 q^{85} +261.247 q^{86} -667.089 q^{89} -958.407 q^{91} -710.996 q^{92} +998.187 q^{94} +327.561 q^{95} -179.654 q^{97} -261.679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 25 q^{5} + 3 q^{7} - 32 q^{8}+O(q^{10}) \) 4 * q - 8 * q^2 + 16 * q^4 - 25 * q^5 + 3 * q^7 - 32 * q^8	\( 4 q - 8 q^{2} + 16 q^{4} - 25 q^{5} + 3 q^{7} - 32 q^{8} + 50 q^{10} - 41 q^{13} - 6 q^{14} + 64 q^{16} - 52 q^{17} + 16 q^{19} - 100 q^{20} - 314 q^{23} - 21 q^{25} + 82 q^{26} + 12 q^{28} - 561 q^{29} + 199 q^{31} - 128 q^{32} + 104 q^{34} + 714 q^{35} + 357 q^{37} - 32 q^{38} + 200 q^{40} - 32 q^{41} - 721 q^{43} + 628 q^{46} - 403 q^{47} + 823 q^{49} + 42 q^{50} - 164 q^{52} + 133 q^{53} - 24 q^{56} + 1122 q^{58} - 1016 q^{59} - 919 q^{61} - 398 q^{62} + 256 q^{64} - 69 q^{65} + 289 q^{67} - 208 q^{68} - 1428 q^{70} + 1205 q^{71} - 1234 q^{73} - 714 q^{74} + 64 q^{76} - 603 q^{79} - 400 q^{80} + 64 q^{82} - 1514 q^{83} + 717 q^{85} + 1442 q^{86} + 1101 q^{89} - 2306 q^{91} - 1256 q^{92} + 806 q^{94} + 1766 q^{95} + 2116 q^{97} - 1646 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 - 25 * q^5 + 3 * q^7 - 32 * q^8 + 50 * q^10 - 41 * q^13 - 6 * q^14 + 64 * q^16 - 52 * q^17 + 16 * q^19 - 100 * q^20 - 314 * q^23 - 21 * q^25 + 82 * q^26 + 12 * q^28 - 561 * q^29 + 199 * q^31 - 128 * q^32 + 104 * q^34 + 714 * q^35 + 357 * q^37 - 32 * q^38 + 200 * q^40 - 32 * q^41 - 721 * q^43 + 628 * q^46 - 403 * q^47 + 823 * q^49 + 42 * q^50 - 164 * q^52 + 133 * q^53 - 24 * q^56 + 1122 * q^58 - 1016 * q^59 - 919 * q^61 - 398 * q^62 + 256 * q^64 - 69 * q^65 + 289 * q^67 - 208 * q^68 - 1428 * q^70 + 1205 * q^71 - 1234 * q^73 - 714 * q^74 + 64 * q^76 - 603 * q^79 - 400 * q^80 + 64 * q^82 - 1514 * q^83 + 717 * q^85 + 1442 * q^86 + 1101 * q^89 - 2306 * q^91 - 1256 * q^92 + 806 * q^94 + 1766 * q^95 + 2116 * q^97 - 1646 * q^98

Introduction

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