LMFDB - Embedded newform 1089.4.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,4,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1089.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:	$64.2530799963$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{5}, \sqrt{37})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 21x^{2} + 64 $ x^4 - 21*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.92335$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.92335 q^{2} +0.545959 q^{4} -6.17671 q^{5} -32.1271 q^{7} -21.7907 q^{8} +O(q^{10})$	$q+2.92335 q^{2} +0.545959 q^{4} -6.17671 q^{5} -32.1271 q^{7} -21.7907 q^{8} -18.0567 q^{10} +4.49714 q^{13} -93.9186 q^{14} -68.0696 q^{16} +59.4763 q^{17} -28.9929 q^{19} -3.37223 q^{20} +38.3004 q^{23} -86.8483 q^{25} +13.1467 q^{26} -17.5401 q^{28} +39.4345 q^{29} -266.057 q^{31} -24.6651 q^{32} +173.870 q^{34} +198.440 q^{35} +112.232 q^{37} -84.7563 q^{38} +134.595 q^{40} -134.223 q^{41} +252.470 q^{43} +111.965 q^{46} +182.276 q^{47} +689.150 q^{49} -253.888 q^{50} +2.45526 q^{52} -42.7271 q^{53} +700.073 q^{56} +115.281 q^{58} +180.626 q^{59} +559.433 q^{61} -777.778 q^{62} +472.452 q^{64} -27.7775 q^{65} +770.935 q^{67} +32.4716 q^{68} +580.108 q^{70} +26.5440 q^{71} -372.155 q^{73} +328.094 q^{74} -15.8289 q^{76} +252.776 q^{79} +420.446 q^{80} -392.380 q^{82} +1055.18 q^{83} -367.368 q^{85} +738.058 q^{86} -58.5392 q^{89} -144.480 q^{91} +20.9105 q^{92} +532.855 q^{94} +179.081 q^{95} -597.668 q^{97} +2014.62 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 14 q^{4} + 11 q^{5} - 25 q^{7} + 66 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 14 * q^4 + 11 * q^5 - 25 * q^7 + 66 * q^8	$ 4 q + 4 q^{2} + 14 q^{4} + 11 q^{5} - 25 q^{7} + 66 q^{8} - 20 q^{10} - 25 q^{13} - 132 q^{14} + 34 q^{16} + 232 q^{17} - 154 q^{19} + 254 q^{20} + 6 q^{23} - 13 q^{25} + 200 q^{26} - 24 q^{28} + 363 q^{29} + 37 q^{31} + 162 q^{32} - 149 q^{34} + 356 q^{35} + 93 q^{37} - 379 q^{38} + 814 q^{40} + 152 q^{41} + 325 q^{43} + 1258 q^{46} + 869 q^{47} - 245 q^{49} - 1016 q^{50} + 1010 q^{52} - 811 q^{53} + 780 q^{56} + 956 q^{58} - 178 q^{59} + 105 q^{61} - 342 q^{62} + 818 q^{64} + 895 q^{65} + 43 q^{67} - 135 q^{68} - 22 q^{70} - 629 q^{71} + 270 q^{73} + 202 q^{74} + 121 q^{76} - 977 q^{79} - 122 q^{80} + 789 q^{82} + 1686 q^{83} + 721 q^{85} + 277 q^{86} - 1891 q^{89} - 80 q^{91} + 3450 q^{92} - 756 q^{94} + 1804 q^{95} - 1772 q^{97} + 1370 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 14 * q^4 + 11 * q^5 - 25 * q^7 + 66 * q^8 - 20 * q^10 - 25 * q^13 - 132 * q^14 + 34 * q^16 + 232 * q^17 - 154 * q^19 + 254 * q^20 + 6 * q^23 - 13 * q^25 + 200 * q^26 - 24 * q^28 + 363 * q^29 + 37 * q^31 + 162 * q^32 - 149 * q^34 + 356 * q^35 + 93 * q^37 - 379 * q^38 + 814 * q^40 + 152 * q^41 + 325 * q^43 + 1258 * q^46 + 869 * q^47 - 245 * q^49 - 1016 * q^50 + 1010 * q^52 - 811 * q^53 + 780 * q^56 + 956 * q^58 - 178 * q^59 + 105 * q^61 - 342 * q^62 + 818 * q^64 + 895 * q^65 + 43 * q^67 - 135 * q^68 - 22 * q^70 - 629 * q^71 + 270 * q^73 + 202 * q^74 + 121 * q^76 - 977 * q^79 - 122 * q^80 + 789 * q^82 + 1686 * q^83 + 721 * q^85 + 277 * q^86 - 1891 * q^89 - 80 * q^91 + 3450 * q^92 - 756 * q^94 + 1804 * q^95 - 1772 * q^97 + 1370 * 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 $

$2$	2.92335	1.03356	0.516780	−	0.856118i	$-0.327131\pi$
$2$	2.92335	1.03356	0.516780	+	0.856118i	$0.327131\pi$
$3$	0	0
$3$	0	0
$4$	0.545959	0.0682449
$5$	−6.17671	−0.552462	−0.276231	−	0.961091i	$-0.589085\pi$
$5$	−6.17671	−0.552462	−0.276231	+	0.961091i	$0.589085\pi$
$6$	0	0
$7$	−32.1271	−1.73470	−0.867350	−	0.497699i	$-0.834178\pi$
$7$	−32.1271	−1.73470	−0.867350	+	0.497699i	$0.834178\pi$
$8$	−21.7907	−0.963024
$9$	0	0
$10$	−18.0567	−0.571002
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.49714	0.0959448	0.0479724	−	0.998849i	$-0.484724\pi$
$13$	4.49714	0.0959448	0.0479724	+	0.998849i	$0.484724\pi$
$14$	−93.9186	−1.79292
$15$	0	0
$16$	−68.0696	−1.06359
$17$	59.4763	0.848536	0.424268	−	0.905537i	$-0.360531\pi$
$17$	59.4763	0.848536	0.424268	+	0.905537i	$0.360531\pi$
$18$	0	0
$19$	−28.9929	−0.350075	−0.175037	−	0.984562i	$-0.556005\pi$
$19$	−28.9929	−0.350075	−0.175037	+	0.984562i	$0.556005\pi$
$20$	−3.37223	−0.0377027
$21$	0	0
$22$	0	0
$23$	38.3004	0.347226	0.173613	−	0.984814i	$-0.444456\pi$
$23$	38.3004	0.347226	0.173613	+	0.984814i	$0.444456\pi$
$24$	0	0
$25$	−86.8483	−0.694786
$26$	13.1467	0.0991647
$27$	0	0
$28$	−17.5401	−0.118384
$29$	39.4345	0.252511	0.126255	−	0.991998i	$-0.459704\pi$
$29$	39.4345	0.252511	0.126255	+	0.991998i	$0.459704\pi$
$30$	0	0
$31$	−266.057	−1.54146	−0.770731	−	0.637161i	$-0.780109\pi$
$31$	−266.057	−1.54146	−0.770731	+	0.637161i	$0.780109\pi$
$32$	−24.6651	−0.136257
$33$	0	0
$34$	173.870	0.877012
$35$	198.440	0.958355
$36$	0	0
$37$	112.232	0.498672	0.249336	−	0.968417i	$-0.419788\pi$
$37$	112.232	0.498672	0.249336	+	0.968417i	$0.419788\pi$
$38$	−84.7563	−0.361823
$39$	0	0
$40$	134.595	0.532034
$41$	−134.223	−0.511271	−0.255635	−	0.966773i	$-0.582285\pi$
$41$	−134.223	−0.511271	−0.255635	+	0.966773i	$0.582285\pi$
$42$	0	0
$43$	252.470	0.895380	0.447690	−	0.894189i	$-0.352247\pi$
$43$	252.470	0.895380	0.447690	+	0.894189i	$0.352247\pi$
$44$	0	0
$45$	0	0
$46$	111.965	0.358878
$47$	182.276	0.565694	0.282847	−	0.959165i	$-0.408721\pi$
$47$	182.276	0.565694	0.282847	+	0.959165i	$0.408721\pi$
$48$	0	0
$49$	689.150	2.00918
$50$	−253.888	−0.718103
$51$	0	0
$52$	2.45526	0.00654775
$53$	−42.7271	−0.110736	−0.0553681	−	0.998466i	$-0.517633\pi$
$53$	−42.7271	−0.110736	−0.0553681	+	0.998466i	$0.517633\pi$
$54$	0	0
$55$	0	0
$56$	700.073	1.67056
$57$	0	0
$58$	115.281	0.260985
$59$	180.626	0.398568	0.199284	−	0.979942i	$-0.436138\pi$
$59$	180.626	0.398568	0.199284	+	0.979942i	$0.436138\pi$
$60$	0	0
$61$	559.433	1.17423	0.587116	−	0.809503i	$-0.300263\pi$
$61$	559.433	1.17423	0.587116	+	0.809503i	$0.300263\pi$
$62$	−777.778	−1.59319
$63$	0	0
$64$	472.452	0.922758
$65$	−27.7775	−0.0530058
$66$	0	0
$67$	770.935	1.40574	0.702871	−	0.711318i	$-0.251901\pi$
$67$	770.935	1.40574	0.702871	+	0.711318i	$0.251901\pi$
$68$	32.4716	0.0579083
$69$	0	0
$70$	580.108	0.990517
$71$	26.5440	0.0443689	0.0221844	−	0.999754i	$-0.492938\pi$
$71$	26.5440	0.0443689	0.0221844	+	0.999754i	$0.492938\pi$
$72$	0	0
$73$	−372.155	−0.596677	−0.298339	−	0.954460i	$-0.596432\pi$
$73$	−372.155	−0.596677	−0.298339	+	0.954460i	$0.596432\pi$
$74$	328.094	0.515407
$75$	0	0
$76$	−15.8289	−0.0238908
$77$	0	0
$78$	0	0
$79$	252.776	0.359994	0.179997	−	0.983667i	$-0.442391\pi$
$79$	252.776	0.359994	0.179997	+	0.983667i	$0.442391\pi$
$80$	420.446	0.587591
$81$	0	0
$82$	−392.380	−0.528429
$83$	1055.18	1.39543	0.697717	−	0.716373i	$-0.254199\pi$
$83$	1055.18	1.39543	0.697717	+	0.716373i	$0.254199\pi$
$84$	0	0
$85$	−367.368	−0.468784
$86$	738.058	0.925429
$87$	0	0
$88$	0	0
$89$	−58.5392	−0.0697206	−0.0348603	−	0.999392i	$-0.511099\pi$
$89$	−58.5392	−0.0697206	−0.0348603	+	0.999392i	$0.511099\pi$
$90$	0	0
$91$	−144.480	−0.166436
$92$	20.9105	0.0236964
$93$	0	0
$94$	532.855	0.584679
$95$	179.081	0.193403
$96$	0	0
$97$	−597.668	−0.625608	−0.312804	−	0.949818i	$-0.601268\pi$
$97$	−597.668	−0.625608	−0.312804	+	0.949818i	$0.601268\pi$
$98$	2014.62	2.07661
$99$	0	0
$100$	−47.4156	−0.0474156
$101$	1280.94	1.26196	0.630981	−	0.775798i	$-0.282653\pi$
$101$	1280.94	1.26196	0.630981	+	0.775798i	$0.282653\pi$
$102$	0	0
$103$	1119.57	1.07102	0.535510	−	0.844529i	$-0.320120\pi$
$103$	1119.57	1.07102	0.535510	+	0.844529i	$0.320120\pi$
$104$	−97.9961	−0.0923972
$105$	0	0
$106$	−124.906	−0.114453
$107$	−2.69158	−0.00243182	−0.00121591	−	0.999999i	$-0.500387\pi$
$107$	−2.69158	−0.00243182	−0.00121591	+	0.999999i	$0.500387\pi$
$108$	0	0
$109$	10.5392	0.00926117	0.00463058	−	0.999989i	$-0.498526\pi$
$109$	10.5392	0.00926117	0.00463058	+	0.999989i	$0.498526\pi$
$110$	0	0
$111$	0	0
$112$	2186.88	1.84501
$113$	−2250.77	−1.87375	−0.936877	−	0.349658i	$-0.886298\pi$
$113$	−2250.77	−1.87375	−0.936877	+	0.349658i	$0.886298\pi$
$114$	0	0
$115$	−236.571	−0.191829
$116$	21.5296	0.0172326
$117$	0	0
$118$	528.033	0.411944
$119$	−1910.80	−1.47196
$120$	0	0
$121$	0	0
$122$	1635.42	1.21364
$123$	0	0
$124$	−145.256	−0.105197
$125$	1308.53	0.936304
$126$	0	0
$127$	−1087.92	−0.760133	−0.380067	−	0.924959i	$-0.624099\pi$
$127$	−1087.92	−0.760133	−0.380067	+	0.924959i	$0.624099\pi$
$128$	1578.46	1.08998
$129$	0	0
$130$	−81.2034	−0.0547847
$131$	−297.144	−0.198180	−0.0990901	−	0.995078i	$-0.531593\pi$
$131$	−297.144	−0.198180	−0.0990901	+	0.995078i	$0.531593\pi$
$132$	0	0
$133$	931.457	0.607275
$134$	2253.71	1.45292
$135$	0	0
$136$	−1296.03	−0.817161
$137$	−2591.05	−1.61583	−0.807915	−	0.589300i	$-0.799404\pi$
$137$	−2591.05	−1.61583	−0.807915	+	0.589300i	$0.799404\pi$
$138$	0	0
$139$	−243.567	−0.148626	−0.0743132	−	0.997235i	$-0.523676\pi$
$139$	−243.567	−0.148626	−0.0743132	+	0.997235i	$0.523676\pi$
$140$	108.340	0.0654029
$141$	0	0
$142$	77.5973	0.0458579
$143$	0	0
$144$	0	0
$145$	−243.575	−0.139502
$146$	−1087.94	−0.616701
$147$	0	0
$148$	61.2743	0.0340319
$149$	−1882.28	−1.03492	−0.517458	−	0.855708i	$-0.673122\pi$
$149$	−1882.28	−1.03492	−0.517458	+	0.855708i	$0.673122\pi$
$150$	0	0
$151$	−1618.58	−0.872305	−0.436153	−	0.899873i	$-0.643659\pi$
$151$	−1618.58	−0.872305	−0.436153	+	0.899873i	$0.643659\pi$
$152$	631.777	0.337131
$153$	0	0
$154$	0	0
$155$	1643.36	0.851598
$156$	0	0
$157$	−1615.16	−0.821045	−0.410523	−	0.911850i	$-0.634654\pi$
$157$	−1615.16	−0.821045	−0.410523	+	0.911850i	$0.634654\pi$
$158$	738.952	0.372075
$159$	0	0
$160$	152.349	0.0752766
$161$	−1230.48	−0.602332
$162$	0	0
$163$	414.211	0.199040	0.0995199	−	0.995036i	$-0.468269\pi$
$163$	414.211	0.199040	0.0995199	+	0.995036i	$0.468269\pi$
$164$	−73.2803	−0.0348916
$165$	0	0
$166$	3084.66	1.44226
$167$	468.013	0.216862	0.108431	−	0.994104i	$-0.465417\pi$
$167$	468.013	0.216862	0.108431	+	0.994104i	$0.465417\pi$
$168$	0	0
$169$	−2176.78	−0.990795
$170$	−1073.94	−0.484516
$171$	0	0
$172$	137.839	0.0611052
$173$	2302.50	1.01188	0.505941	−	0.862568i	$-0.331145\pi$
$173$	2302.50	1.01188	0.505941	+	0.862568i	$0.331145\pi$
$174$	0	0
$175$	2790.18	1.20525
$176$	0	0
$177$	0	0
$178$	−171.130	−0.0720604
$179$	3381.94	1.41217	0.706083	−	0.708129i	$-0.250460\pi$
$179$	3381.94	1.41217	0.706083	+	0.708129i	$0.250460\pi$
$180$	0	0
$181$	−3675.98	−1.50958	−0.754789	−	0.655968i	$-0.772261\pi$
$181$	−3675.98	−1.50958	−0.754789	+	0.655968i	$0.772261\pi$
$182$	−422.366	−0.172021
$183$	0	0
$184$	−834.595	−0.334387
$185$	−693.226	−0.275497
$186$	0	0
$187$	0	0
$188$	99.5151	0.0386058
$189$	0	0
$190$	523.515	0.199893
$191$	1778.66	0.673819	0.336910	−	0.941537i	$-0.390618\pi$
$191$	1778.66	0.673819	0.336910	+	0.941537i	$0.390618\pi$
$192$	0	0
$193$	3026.33	1.12870	0.564352	−	0.825534i	$-0.309126\pi$
$193$	3026.33	1.12870	0.564352	+	0.825534i	$0.309126\pi$
$194$	−1747.19	−0.646603
$195$	0	0
$196$	376.248	0.137117
$197$	4415.89	1.59705	0.798527	−	0.601960i	$-0.205613\pi$
$197$	4415.89	1.59705	0.798527	+	0.601960i	$0.205613\pi$
$198$	0	0
$199$	−4105.13	−1.46234	−0.731169	−	0.682196i	$-0.761025\pi$
$199$	−4105.13	−1.46234	−0.731169	+	0.682196i	$0.761025\pi$
$200$	1892.49	0.669096
$201$	0	0
$202$	3744.63	1.30431
$203$	−1266.92	−0.438030
$204$	0	0
$205$	829.056	0.282458
$206$	3272.91	1.10696
$207$	0	0
$208$	−306.119	−0.102046
$209$	0	0
$210$	0	0
$211$	−2851.03	−0.930204	−0.465102	−	0.885257i	$-0.653982\pi$
$211$	−2851.03	−0.930204	−0.465102	+	0.885257i	$0.653982\pi$
$212$	−23.3273	−0.00755719
$213$	0	0
$214$	−7.86841	−0.00251343
$215$	−1559.44	−0.494663
$216$	0	0
$217$	8547.65	2.67397
$218$	30.8096	0.00957197
$219$	0	0
$220$	0	0
$221$	267.473	0.0814127
$222$	0	0
$223$	2110.98	0.633910	0.316955	−	0.948441i	$-0.397340\pi$
$223$	2110.98	0.633910	0.316955	+	0.948441i	$0.397340\pi$
$224$	792.418	0.236364
$225$	0	0
$226$	−6579.77	−1.93664
$227$	1118.76	0.327113	0.163556	−	0.986534i	$-0.447703\pi$
$227$	1118.76	0.327113	0.163556	+	0.986534i	$0.447703\pi$
$228$	0	0
$229$	852.481	0.245998	0.122999	−	0.992407i	$-0.460749\pi$
$229$	852.481	0.245998	0.122999	+	0.992407i	$0.460749\pi$
$230$	−691.578	−0.198267
$231$	0	0
$232$	−859.308	−0.243174
$233$	−3158.86	−0.888171	−0.444086	−	0.895984i	$-0.646471\pi$
$233$	−3158.86	−0.888171	−0.444086	+	0.895984i	$0.646471\pi$
$234$	0	0
$235$	−1125.86	−0.312524
$236$	98.6145	0.0272002
$237$	0	0
$238$	−5585.93	−1.52135
$239$	4442.88	1.20245	0.601226	−	0.799079i	$-0.294679\pi$
$239$	4442.88	1.20245	0.601226	+	0.799079i	$0.294679\pi$
$240$	0	0
$241$	1903.11	0.508673	0.254336	−	0.967116i	$-0.418143\pi$
$241$	1903.11	0.508673	0.254336	+	0.967116i	$0.418143\pi$
$242$	0	0
$243$	0	0
$244$	305.428	0.0801353
$245$	−4256.68	−1.11000
$246$	0	0
$247$	−130.385	−0.0335879
$248$	5797.59	1.48446
$249$	0	0
$250$	3825.27	0.967726
$251$	1790.11	0.450162	0.225081	−	0.974340i	$-0.427735\pi$
$251$	1790.11	0.450162	0.225081	+	0.974340i	$0.427735\pi$
$252$	0	0
$253$	0	0
$254$	−3180.36	−0.785643
$255$	0	0
$256$	834.777	0.203803
$257$	−5759.65	−1.39797	−0.698983	−	0.715138i	$-0.746364\pi$
$257$	−5759.65	−1.39797	−0.698983	+	0.715138i	$0.746364\pi$
$258$	0	0
$259$	−3605.70	−0.865047
$260$	−15.1654	−0.00361738
$261$	0	0
$262$	−868.656	−0.204831
$263$	5675.79	1.33074	0.665370	−	0.746514i	$-0.268274\pi$
$263$	5675.79	1.33074	0.665370	+	0.746514i	$0.268274\pi$
$264$	0	0
$265$	263.913	0.0611775
$266$	2722.97	0.627655
$267$	0	0
$268$	420.899	0.0959347
$269$	1628.03	0.369006	0.184503	−	0.982832i	$-0.440932\pi$
$269$	1628.03	0.369006	0.184503	+	0.982832i	$0.440932\pi$
$270$	0	0
$271$	5533.83	1.24043	0.620215	−	0.784432i	$-0.287045\pi$
$271$	5533.83	1.24043	0.620215	+	0.784432i	$0.287045\pi$
$272$	−4048.53	−0.902492
$273$	0	0
$274$	−7574.55	−1.67006
$275$	0	0
$276$	0	0
$277$	5357.23	1.16204	0.581020	−	0.813889i	$-0.302654\pi$
$277$	5357.23	1.16204	0.581020	+	0.813889i	$0.302654\pi$
$278$	−712.030	−0.153614
$279$	0	0
$280$	−4324.15	−0.922919
$281$	−5723.51	−1.21508	−0.607538	−	0.794291i	$-0.707843\pi$
$281$	−5723.51	−1.21508	−0.607538	+	0.794291i	$0.707843\pi$
$282$	0	0
$283$	−5837.12	−1.22608	−0.613041	−	0.790051i	$-0.710054\pi$
$283$	−5837.12	−1.22608	−0.613041	+	0.790051i	$0.710054\pi$
$284$	14.4919	0.00302795
$285$	0	0
$286$	0	0
$287$	4312.19	0.886902
$288$	0	0
$289$	−1375.57	−0.279987
$290$	−712.056	−0.144184
$291$	0	0
$292$	−203.181	−0.0407202
$293$	−1414.09	−0.281952	−0.140976	−	0.990013i	$-0.545024\pi$
$293$	−1414.09	−0.281952	−0.140976	+	0.990013i	$0.545024\pi$
$294$	0	0
$295$	−1115.67	−0.220194
$296$	−2445.63	−0.480234
$297$	0	0
$298$	−5502.56	−1.06965
$299$	172.243	0.0333145
$300$	0	0
$301$	−8111.14	−1.55322
$302$	−4731.67	−0.901579
$303$	0	0
$304$	1973.53	0.372335
$305$	−3455.46	−0.648718
$306$	0	0
$307$	6080.13	1.13033	0.565165	−	0.824978i	$-0.308812\pi$
$307$	6080.13	1.13033	0.565165	+	0.824978i	$0.308812\pi$
$308$	0	0
$309$	0	0
$310$	4804.11	0.880177
$311$	6572.40	1.19835	0.599174	−	0.800619i	$-0.295496\pi$
$311$	6572.40	1.19835	0.599174	+	0.800619i	$0.295496\pi$
$312$	0	0
$313$	−2977.42	−0.537680	−0.268840	−	0.963185i	$-0.586640\pi$
$313$	−2977.42	−0.537680	−0.268840	+	0.963185i	$0.586640\pi$
$314$	−4721.68	−0.848599
$315$	0	0
$316$	138.005	0.0245678
$317$	−2239.07	−0.396715	−0.198357	−	0.980130i	$-0.563561\pi$
$317$	−2239.07	−0.396715	−0.198357	+	0.980130i	$0.563561\pi$
$318$	0	0
$319$	0	0
$320$	−2918.20	−0.509788
$321$	0	0
$322$	−3597.12	−0.622546
$323$	−1724.39	−0.297051
$324$	0	0
$325$	−390.569	−0.0666612
$326$	1210.88	0.205720
$327$	0	0
$328$	2924.82	0.492366
$329$	−5855.99	−0.981310
$330$	0	0
$331$	−1551.70	−0.257671	−0.128835	−	0.991666i	$-0.541124\pi$
$331$	−1551.70	−0.257671	−0.128835	+	0.991666i	$0.541124\pi$
$332$	576.085	0.0952313
$333$	0	0
$334$	1368.16	0.224139
$335$	−4761.84	−0.776618
$336$	0	0
$337$	6801.54	1.09942	0.549708	−	0.835357i	$-0.314739\pi$
$337$	6801.54	1.09942	0.549708	+	0.835357i	$0.314739\pi$
$338$	−6363.47	−1.02404
$339$	0	0
$340$	−200.568	−0.0319921
$341$	0	0
$342$	0	0
$343$	−11120.8	−1.75063
$344$	−5501.52	−0.862273
$345$	0	0
$346$	6731.00	1.04584
$347$	8299.68	1.28401	0.642003	−	0.766702i	$-0.278104\pi$
$347$	8299.68	1.28401	0.642003	+	0.766702i	$0.278104\pi$
$348$	0	0
$349$	−64.3320	−0.00986708	−0.00493354	−	0.999988i	$-0.501570\pi$
$349$	−64.3320	−0.00986708	−0.00493354	+	0.999988i	$0.501570\pi$
$350$	8156.67	1.24569
$351$	0	0
$352$	0	0
$353$	−5757.22	−0.868062	−0.434031	−	0.900898i	$-0.642909\pi$
$353$	−5757.22	−0.868062	−0.434031	+	0.900898i	$0.642909\pi$
$354$	0	0
$355$	−163.954	−0.0245121
$356$	−31.9600	−0.00475808
$357$	0	0
$358$	9886.57	1.45956
$359$	12590.2	1.85093	0.925465	−	0.378833i	$-0.123674\pi$
$359$	12590.2	1.85093	0.925465	+	0.378833i	$0.123674\pi$
$360$	0	0
$361$	−6018.41	−0.877448
$362$	−10746.2	−1.56024
$363$	0	0
$364$	−78.8803	−0.0113584
$365$	2298.69	0.329641
$366$	0	0
$367$	7662.22	1.08982	0.544911	−	0.838494i	$-0.316564\pi$
$367$	7662.22	1.08982	0.544911	+	0.838494i	$0.316564\pi$
$368$	−2607.10	−0.369305
$369$	0	0
$370$	−2026.54	−0.284743
$371$	1372.70	0.192094
$372$	0	0
$373$	7421.71	1.03024	0.515122	−	0.857117i	$-0.327746\pi$
$373$	7421.71	1.03024	0.515122	+	0.857117i	$0.327746\pi$
$374$	0	0
$375$	0	0
$376$	−3971.92	−0.544777
$377$	177.343	0.0242271
$378$	0	0
$379$	5526.10	0.748962	0.374481	−	0.927235i	$-0.377821\pi$
$379$	5526.10	0.748962	0.374481	+	0.927235i	$0.377821\pi$
$380$	97.7707	0.0131988
$381$	0	0
$382$	5199.65	0.696432
$383$	−9043.90	−1.20658	−0.603292	−	0.797520i	$-0.706145\pi$
$383$	−9043.90	−1.20658	−0.603292	+	0.797520i	$0.706145\pi$
$384$	0	0
$385$	0	0
$386$	8847.01	1.16658
$387$	0	0
$388$	−326.302	−0.0426946
$389$	11777.7	1.53510	0.767552	−	0.640987i	$-0.221475\pi$
$389$	11777.7	1.53510	0.767552	+	0.640987i	$0.221475\pi$
$390$	0	0
$391$	2277.97	0.294634
$392$	−15017.1	−1.93489
$393$	0	0
$394$	12909.2	1.65065
$395$	−1561.32	−0.198883
$396$	0	0
$397$	8173.72	1.03332	0.516659	−	0.856191i	$-0.327176\pi$
$397$	8173.72	1.03332	0.516659	+	0.856191i	$0.327176\pi$
$398$	−12000.7	−1.51141
$399$	0	0
$400$	5911.73	0.738966
$401$	8666.14	1.07922	0.539609	−	0.841916i	$-0.318572\pi$
$401$	8666.14	1.07922	0.539609	+	0.841916i	$0.318572\pi$
$402$	0	0
$403$	−1196.50	−0.147895
$404$	699.340	0.0861225
$405$	0	0
$406$	−3703.64	−0.452730
$407$	0	0
$408$	0	0
$409$	−2320.05	−0.280486	−0.140243	−	0.990117i	$-0.544788\pi$
$409$	−2320.05	−0.280486	−0.140243	+	0.990117i	$0.544788\pi$
$410$	2423.62	0.291937
$411$	0	0
$412$	611.242	0.0730916
$413$	−5802.99	−0.691396
$414$	0	0
$415$	−6517.54	−0.770924
$416$	−110.922	−0.0130731
$417$	0	0
$418$	0	0
$419$	−12620.5	−1.47148	−0.735740	−	0.677264i	$-0.763165\pi$
$419$	−12620.5	−1.47148	−0.735740	+	0.677264i	$0.763165\pi$
$420$	0	0
$421$	8305.13	0.961442	0.480721	−	0.876873i	$-0.340375\pi$
$421$	8305.13	0.961442	0.480721	+	0.876873i	$0.340375\pi$
$422$	−8334.55	−0.961421
$423$	0	0
$424$	931.056	0.106642
$425$	−5165.41	−0.589551
$426$	0	0
$427$	−17973.0	−2.03694
$428$	−1.46949	−0.000165959 0
$429$	0	0
$430$	−4558.77	−0.511264
$431$	−10156.2	−1.13505	−0.567526	−	0.823356i	$-0.692099\pi$
$431$	−10156.2	−1.13505	−0.567526	+	0.823356i	$0.692099\pi$
$432$	0	0
$433$	−1944.71	−0.215836	−0.107918	−	0.994160i	$-0.534418\pi$
$433$	−1944.71	−0.215836	−0.107918	+	0.994160i	$0.534418\pi$
$434$	24987.7	2.76371
$435$	0	0
$436$	5.75395	0.000632028 0
$437$	−1110.44	−0.121555
$438$	0	0
$439$	−15398.2	−1.67407	−0.837035	−	0.547150i	$-0.815713\pi$
$439$	−15398.2	−1.67407	−0.837035	+	0.547150i	$0.815713\pi$
$440$	0	0
$441$	0	0
$442$	781.917	0.0841448
$443$	13035.8	1.39808	0.699039	−	0.715083i	$-0.253611\pi$
$443$	13035.8	1.39808	0.699039	+	0.715083i	$0.253611\pi$
$444$	0	0
$445$	361.579	0.0385180
$446$	6171.14	0.655183
$447$	0	0
$448$	−15178.5	−1.60071
$449$	7783.29	0.818076	0.409038	−	0.912517i	$-0.365864\pi$
$449$	7783.29	0.818076	0.409038	+	0.912517i	$0.365864\pi$
$450$	0	0
$451$	0	0
$452$	−1228.83	−0.127874
$453$	0	0
$454$	3270.52	0.338090
$455$	892.412	0.0919492
$456$	0	0
$457$	505.825	0.0517757	0.0258878	−	0.999665i	$-0.491759\pi$
$457$	505.825	0.0517757	0.0258878	+	0.999665i	$0.491759\pi$
$458$	2492.10	0.254254
$459$	0	0
$460$	−129.158	−0.0130913
$461$	9973.13	1.00758	0.503791	−	0.863826i	$-0.331938\pi$
$461$	9973.13	1.00758	0.503791	+	0.863826i	$0.331938\pi$
$462$	0	0
$463$	5539.97	0.556078	0.278039	−	0.960570i	$-0.410315\pi$
$463$	5539.97	0.556078	0.278039	+	0.960570i	$0.410315\pi$
$464$	−2684.29	−0.268567
$465$	0	0
$466$	−9234.45	−0.917978
$467$	15646.4	1.55039	0.775194	−	0.631723i	$-0.217652\pi$
$467$	15646.4	1.55039	0.775194	+	0.631723i	$0.217652\pi$
$468$	0	0
$469$	−24767.9	−2.43854
$470$	−3291.29	−0.323013
$471$	0	0
$472$	−3935.98	−0.383831
$473$	0	0
$474$	0	0
$475$	2517.98	0.243227
$476$	−1043.22	−0.100453
$477$	0	0
$478$	12988.1	1.24281
$479$	15820.0	1.50905	0.754523	−	0.656273i	$-0.227868\pi$
$479$	15820.0	1.50905	0.754523	+	0.656273i	$0.227868\pi$
$480$	0	0
$481$	504.725	0.0478450
$482$	5563.45	0.525743
$483$	0	0
$484$	0	0
$485$	3691.62	0.345625
$486$	0	0
$487$	−1035.46	−0.0963475	−0.0481738	−	0.998839i	$-0.515340\pi$
$487$	−1035.46	−0.0963475	−0.0481738	+	0.998839i	$0.515340\pi$
$488$	−12190.5	−1.13081
$489$	0	0
$490$	−12443.7	−1.14725
$491$	9385.84	0.862682	0.431341	−	0.902189i	$-0.358041\pi$
$491$	9385.84	0.862682	0.431341	+	0.902189i	$0.358041\pi$
$492$	0	0
$493$	2345.42	0.214264
$494$	−381.161	−0.0347151
$495$	0	0
$496$	18110.4	1.63948
$497$	−852.781	−0.0769667
$498$	0	0
$499$	−4640.55	−0.416312	−0.208156	−	0.978096i	$-0.566746\pi$
$499$	−4640.55	−0.416312	−0.208156	+	0.978096i	$0.566746\pi$
$500$	714.401	0.0638980
$501$	0	0
$502$	5233.11	0.465270
$503$	−4183.00	−0.370797	−0.185398	−	0.982663i	$-0.559358\pi$
$503$	−4183.00	−0.370797	−0.185398	+	0.982663i	$0.559358\pi$
$504$	0	0
$505$	−7911.98	−0.697186
$506$	0	0
$507$	0	0
$508$	−593.958	−0.0518752
$509$	−17980.0	−1.56572	−0.782860	−	0.622198i	$-0.786240\pi$
$509$	−17980.0	−1.56572	−0.782860	+	0.622198i	$0.786240\pi$
$510$	0	0
$511$	11956.3	1.03506
$512$	−10187.4	−0.879340
$513$	0	0
$514$	−16837.5	−1.44488
$515$	−6915.29	−0.591697
$516$	0	0
$517$	0	0
$518$	−10540.7	−0.894077
$519$	0	0
$520$	605.293	0.0510459
$521$	−8211.54	−0.690507	−0.345253	−	0.938510i	$-0.612207\pi$
$521$	−8211.54	−0.690507	−0.345253	+	0.938510i	$0.612207\pi$
$522$	0	0
$523$	13820.0	1.15546	0.577729	−	0.816228i	$-0.303939\pi$
$523$	13820.0	1.15546	0.577729	+	0.816228i	$0.303939\pi$
$524$	−162.229	−0.0135248
$525$	0	0
$526$	16592.3	1.37540
$527$	−15824.1	−1.30799
$528$	0	0
$529$	−10700.1	−0.879434
$530$	771.509	0.0632306
$531$	0	0
$532$	508.538	0.0414434
$533$	−603.620	−0.0490538
$534$	0	0
$535$	16.6251	0.00134349
$536$	−16799.2	−1.35376
$537$	0	0
$538$	4759.29	0.381390
$539$	0	0
$540$	0	0
$541$	3747.36	0.297803	0.148902	−	0.988852i	$-0.452426\pi$
$541$	3747.36	0.297803	0.148902	+	0.988852i	$0.452426\pi$
$542$	16177.3	1.28206
$543$	0	0
$544$	−1466.99	−0.115619
$545$	−65.0973	−0.00511644
$546$	0	0
$547$	12440.2	0.972400	0.486200	−	0.873847i	$-0.338383\pi$
$547$	12440.2	0.972400	0.486200	+	0.873847i	$0.338383\pi$
$548$	−1414.61	−0.110272
$549$	0	0
$550$	0	0
$551$	−1143.32	−0.0883976
$552$	0	0
$553$	−8120.96	−0.624482
$554$	15661.1	1.20104
$555$	0	0
$556$	−132.978	−0.0101430
$557$	6386.64	0.485836	0.242918	−	0.970047i	$-0.421895\pi$
$557$	6386.64	0.485836	0.242918	+	0.970047i	$0.421895\pi$
$558$	0	0
$559$	1135.40	0.0859071
$560$	−13507.7	−1.01929
$561$	0	0
$562$	−16731.8	−1.25585
$563$	−13966.9	−1.04553	−0.522765	−	0.852477i	$-0.675099\pi$
$563$	−13966.9	−1.04553	−0.522765	+	0.852477i	$0.675099\pi$
$564$	0	0
$565$	13902.3	1.03518
$566$	−17063.9	−1.26723
$567$	0	0
$568$	−578.413	−0.0427283
$569$	19058.3	1.40416	0.702078	−	0.712100i	$-0.252256\pi$
$569$	19058.3	1.40416	0.702078	+	0.712100i	$0.252256\pi$
$570$	0	0
$571$	6805.90	0.498806	0.249403	−	0.968400i	$-0.419766\pi$
$571$	6805.90	0.498806	0.249403	+	0.968400i	$0.419766\pi$
$572$	0	0
$573$	0	0
$574$	12606.0	0.916666
$575$	−3326.33	−0.241248
$576$	0	0
$577$	11014.4	0.794689	0.397344	−	0.917670i	$-0.369932\pi$
$577$	11014.4	0.794689	0.397344	+	0.917670i	$0.369932\pi$
$578$	−4021.28	−0.289383
$579$	0	0
$580$	−132.982	−0.00952033
$581$	−33899.9	−2.42066
$582$	0	0
$583$	0	0
$584$	8109.53	0.574615
$585$	0	0
$586$	−4133.87	−0.291414
$587$	4754.72	0.334324	0.167162	−	0.985929i	$-0.446540\pi$
$587$	4754.72	0.334324	0.167162	+	0.985929i	$0.446540\pi$
$588$	0	0
$589$	7713.77	0.539627
$590$	−3261.50	−0.227583
$591$	0	0
$592$	−7639.61	−0.530382
$593$	8479.59	0.587209	0.293604	−	0.955927i	$-0.405145\pi$
$593$	8479.59	0.587209	0.293604	+	0.955927i	$0.405145\pi$
$594$	0	0
$595$	11802.5	0.813199
$596$	−1027.65	−0.0706278
$597$	0	0
$598$	503.525	0.0344325
$599$	7838.58	0.534684	0.267342	−	0.963602i	$-0.413855\pi$
$599$	7838.58	0.534684	0.267342	+	0.963602i	$0.413855\pi$
$600$	0	0
$601$	27421.1	1.86112	0.930558	−	0.366146i	$-0.119323\pi$
$601$	27421.1	1.86112	0.930558	+	0.366146i	$0.119323\pi$
$602$	−23711.7	−1.60534
$603$	0	0
$604$	−883.678	−0.0595304
$605$	0	0
$606$	0	0
$607$	−24778.0	−1.65685	−0.828423	−	0.560102i	$-0.810762\pi$
$607$	−24778.0	−1.65685	−0.828423	+	0.560102i	$0.810762\pi$
$608$	715.112	0.0477000
$609$	0	0
$610$	−10101.5	−0.670488
$611$	819.720	0.0542755
$612$	0	0
$613$	−15133.1	−0.997098	−0.498549	−	0.866862i	$-0.666134\pi$
$613$	−15133.1	−0.997098	−0.498549	+	0.866862i	$0.666134\pi$
$614$	17774.3	1.16826
$615$	0	0
$616$	0	0
$617$	−15497.8	−1.01121	−0.505605	−	0.862765i	$-0.668731\pi$
$617$	−15497.8	−1.01121	−0.505605	+	0.862765i	$0.668731\pi$
$618$	0	0
$619$	−9208.55	−0.597937	−0.298969	−	0.954263i	$-0.596643\pi$
$619$	−9208.55	−0.597937	−0.298969	+	0.954263i	$0.596643\pi$
$620$	897.207	0.0581173
$621$	0	0
$622$	19213.4	1.23856
$623$	1880.69	0.120944
$624$	0	0
$625$	2773.66	0.177514
$626$	−8704.03	−0.555724
$627$	0	0
$628$	−881.814	−0.0560322
$629$	6675.16	0.423141
$630$	0	0
$631$	3182.55	0.200785	0.100393	−	0.994948i	$-0.467990\pi$
$631$	3182.55	0.200785	0.100393	+	0.994948i	$0.467990\pi$
$632$	−5508.18	−0.346683
$633$	0	0
$634$	−6545.57	−0.410028
$635$	6719.74	0.419944
$636$	0	0
$637$	3099.21	0.192771
$638$	0	0
$639$	0	0
$640$	−9749.70	−0.602173
$641$	2153.33	0.132686	0.0663428	−	0.997797i	$-0.478867\pi$
$641$	2153.33	0.132686	0.0663428	+	0.997797i	$0.478867\pi$
$642$	0	0
$643$	−23264.7	−1.42686	−0.713429	−	0.700727i	$-0.752859\pi$
$643$	−23264.7	−1.42686	−0.713429	+	0.700727i	$0.752859\pi$
$644$	−671.793	−0.0411061
$645$	0	0
$646$	−5040.99	−0.307020
$647$	14758.9	0.896806	0.448403	−	0.893832i	$-0.351993\pi$
$647$	14758.9	0.896806	0.448403	+	0.893832i	$0.351993\pi$
$648$	0	0
$649$	0	0
$650$	−1141.77	−0.0688983
$651$	0	0
$652$	226.142	0.0135835
$653$	3774.08	0.226173	0.113087	−	0.993585i	$-0.463926\pi$
$653$	3774.08	0.226173	0.113087	+	0.993585i	$0.463926\pi$
$654$	0	0
$655$	1835.37	0.109487
$656$	9136.51	0.543781
$657$	0	0
$658$	−17119.1	−1.01424
$659$	−9973.47	−0.589547	−0.294773	−	0.955567i	$-0.595244\pi$
$659$	−9973.47	−0.589547	−0.294773	+	0.955567i	$0.595244\pi$
$660$	0	0
$661$	−24856.8	−1.46266	−0.731329	−	0.682025i	$-0.761100\pi$
$661$	−24856.8	−1.46266	−0.731329	+	0.682025i	$0.761100\pi$
$662$	−4536.15	−0.266318
$663$	0	0
$664$	−22993.2	−1.34384
$665$	−5753.34	−0.335496
$666$	0	0
$667$	1510.36	0.0876781
$668$	255.516	0.0147997
$669$	0	0
$670$	−13920.5	−0.802681
$671$	0	0
$672$	0	0
$673$	14191.4	0.812834	0.406417	−	0.913688i	$-0.366778\pi$
$673$	14191.4	0.812834	0.406417	+	0.913688i	$0.366778\pi$
$674$	19883.3	1.13631
$675$	0	0
$676$	−1188.43	−0.0676167
$677$	12041.6	0.683599	0.341799	−	0.939773i	$-0.388964\pi$
$677$	12041.6	0.683599	0.341799	+	0.939773i	$0.388964\pi$
$678$	0	0
$679$	19201.3	1.08524
$680$	8005.21	0.451450
$681$	0	0
$682$	0	0
$683$	−3428.81	−0.192093	−0.0960467	−	0.995377i	$-0.530620\pi$
$683$	−3428.81	−0.192093	−0.0960467	+	0.995377i	$0.530620\pi$
$684$	0	0
$685$	16004.2	0.892684
$686$	−32509.9	−1.80938
$687$	0	0
$688$	−17185.6	−0.952316
$689$	−192.150	−0.0106246
$690$	0	0
$691$	26382.2	1.45243	0.726213	−	0.687469i	$-0.241278\pi$
$691$	26382.2	1.45243	0.726213	+	0.687469i	$0.241278\pi$
$692$	1257.07	0.0690558
$693$	0	0
$694$	24262.8	1.32710
$695$	1504.44	0.0821104
$696$	0	0
$697$	−7983.08	−0.433832
$698$	−188.065	−0.0101982
$699$	0	0
$700$	1523.33	0.0822519
$701$	−3052.43	−0.164463	−0.0822317	−	0.996613i	$-0.526205\pi$
$701$	−3052.43	−0.164463	−0.0822317	+	0.996613i	$0.526205\pi$
$702$	0	0
$703$	−3253.94	−0.174573
$704$	0	0
$705$	0	0
$706$	−16830.4	−0.897194
$707$	−41152.8	−2.18913
$708$	0	0
$709$	−6702.32	−0.355022	−0.177511	−	0.984119i	$-0.556805\pi$
$709$	−6702.32	−0.355022	−0.177511	+	0.984119i	$0.556805\pi$
$710$	−479.296	−0.0253347
$711$	0	0
$712$	1275.61	0.0671427
$713$	−10190.1	−0.535235
$714$	0	0
$715$	0	0
$716$	1846.40	0.0963732
$717$	0	0
$718$	36805.5	1.91305
$719$	16564.6	0.859188	0.429594	−	0.903022i	$-0.358657\pi$
$719$	16564.6	0.859188	0.429594	+	0.903022i	$0.358657\pi$
$720$	0	0
$721$	−35968.7	−1.85790
$722$	−17593.9	−0.906894
$723$	0	0
$724$	−2006.94	−0.103021
$725$	−3424.82	−0.175441
$726$	0	0
$727$	−26489.8	−1.35138	−0.675689	−	0.737187i	$-0.736154\pi$
$727$	−26489.8	−1.35138	−0.675689	+	0.737187i	$0.736154\pi$
$728$	3148.33	0.160281
$729$	0	0
$730$	6719.88	0.340704
$731$	15016.0	0.759763
$732$	0	0
$733$	12401.8	0.624924	0.312462	−	0.949930i	$-0.398846\pi$
$733$	12401.8	0.624924	0.312462	+	0.949930i	$0.398846\pi$
$734$	22399.3	1.12640
$735$	0	0
$736$	−944.684	−0.0473118
$737$	0	0
$738$	0	0
$739$	15374.2	0.765291	0.382645	−	0.923895i	$-0.375013\pi$
$739$	15374.2	0.765291	0.382645	+	0.923895i	$0.375013\pi$
$740$	−378.473	−0.0188013
$741$	0	0
$742$	4012.87	0.198541
$743$	−13918.4	−0.687238	−0.343619	−	0.939109i	$-0.611653\pi$
$743$	−13918.4	−0.687238	−0.343619	+	0.939109i	$0.611653\pi$
$744$	0	0
$745$	11626.3	0.571752
$746$	21696.2	1.06482
$747$	0	0
$748$	0	0
$749$	86.4725	0.00421847
$750$	0	0
$751$	16460.6	0.799806	0.399903	−	0.916557i	$-0.369044\pi$
$751$	16460.6	0.799806	0.399903	+	0.916557i	$0.369044\pi$
$752$	−12407.4	−0.601666
$753$	0	0
$754$	518.434	0.0250401
$755$	9997.49	0.481915
$756$	0	0
$757$	−19863.9	−0.953718	−0.476859	−	0.878980i	$-0.658225\pi$
$757$	−19863.9	−0.953718	−0.476859	+	0.878980i	$0.658225\pi$
$758$	16154.7	0.774096
$759$	0	0
$760$	−3902.30	−0.186252
$761$	860.397	0.0409847	0.0204924	−	0.999790i	$-0.493477\pi$
$761$	860.397	0.0409847	0.0204924	+	0.999790i	$0.493477\pi$
$762$	0	0
$763$	−338.592	−0.0160653
$764$	971.077	0.0459847
$765$	0	0
$766$	−26438.5	−1.24708
$767$	812.301	0.0382406
$768$	0	0
$769$	18337.1	0.859885	0.429942	−	0.902856i	$-0.358534\pi$
$769$	18337.1	0.859885	0.429942	+	0.902856i	$0.358534\pi$
$770$	0	0
$771$	0	0
$772$	1652.25	0.0770283
$773$	40910.3	1.90354	0.951772	−	0.306806i	$-0.0992604\pi$
$773$	40910.3	1.90354	0.951772	+	0.306806i	$0.0992604\pi$
$774$	0	0
$775$	23106.6	1.07099
$776$	13023.6	0.602476
$777$	0	0
$778$	34430.4	1.58662
$779$	3891.51	0.178983
$780$	0	0
$781$	0	0
$782$	6659.29	0.304521
$783$	0	0
$784$	−46910.2	−2.13694
$785$	9976.39	0.453596
$786$	0	0
$787$	−33328.3	−1.50956	−0.754781	−	0.655977i	$-0.772257\pi$
$787$	−33328.3	−1.50956	−0.754781	+	0.655977i	$0.772257\pi$
$788$	2410.90	0.108991
$789$	0	0
$790$	−4564.29	−0.205557
$791$	72310.6	3.25040
$792$	0	0
$793$	2515.85	0.112661
$794$	23894.6	1.06799
$795$	0	0
$796$	−2241.24	−0.0997971
$797$	11904.7	0.529090	0.264545	−	0.964373i	$-0.414778\pi$
$797$	11904.7	0.529090	0.264545	+	0.964373i	$0.414778\pi$
$798$	0	0
$799$	10841.1	0.480012
$800$	2142.12	0.0946692
$801$	0	0
$802$	25334.2	1.11544
$803$	0	0
$804$	0	0
$805$	7600.32	0.332766
$806$	−3497.78	−0.152859
$807$	0	0
$808$	−27912.6	−1.21530
$809$	−43695.8	−1.89897	−0.949483	−	0.313818i	$-0.898392\pi$
$809$	−43695.8	−1.89897	−0.949483	+	0.313818i	$0.898392\pi$
$810$	0	0
$811$	32650.0	1.41368	0.706842	−	0.707371i	$-0.250119\pi$
$811$	32650.0	1.41368	0.706842	+	0.707371i	$0.250119\pi$
$812$	−691.685	−0.0298933
$813$	0	0
$814$	0	0
$815$	−2558.46	−0.109962
$816$	0	0
$817$	−7319.84	−0.313450
$818$	−6782.30	−0.289899
$819$	0	0
$820$	452.631	0.0192763
$821$	−16030.9	−0.681464	−0.340732	−	0.940160i	$-0.610675\pi$
$821$	−16030.9	−0.681464	−0.340732	+	0.940160i	$0.610675\pi$
$822$	0	0
$823$	−15092.4	−0.639232	−0.319616	−	0.947547i	$-0.603554\pi$
$823$	−15092.4	−0.639232	−0.319616	+	0.947547i	$0.603554\pi$
$824$	−24396.4	−1.03142
$825$	0	0
$826$	−16964.2	−0.714599
$827$	−11534.5	−0.485000	−0.242500	−	0.970151i	$-0.577967\pi$
$827$	−11534.5	−0.485000	−0.242500	+	0.970151i	$0.577967\pi$
$828$	0	0
$829$	39259.4	1.64479	0.822397	−	0.568915i	$-0.192636\pi$
$829$	39259.4	1.64479	0.822397	+	0.568915i	$0.192636\pi$
$830$	−19053.0	−0.796796
$831$	0	0
$832$	2124.69	0.0885339
$833$	40988.1	1.70486
$834$	0	0
$835$	−2890.78	−0.119808
$836$	0	0
$837$	0	0
$838$	−36894.0	−1.52086
$839$	−23334.3	−0.960180	−0.480090	−	0.877219i	$-0.659396\pi$
$839$	−23334.3	−0.960180	−0.480090	+	0.877219i	$0.659396\pi$
$840$	0	0
$841$	−22833.9	−0.936238
$842$	24278.8	0.993708
$843$	0	0
$844$	−1556.55	−0.0634817
$845$	13445.3	0.547376
$846$	0	0
$847$	0	0
$848$	2908.42	0.117778
$849$	0	0
$850$	−15100.3	−0.609336
$851$	4298.55	0.173152
$852$	0	0
$853$	2585.70	0.103790	0.0518949	−	0.998653i	$-0.483474\pi$
$853$	2585.70	0.103790	0.0518949	+	0.998653i	$0.483474\pi$
$854$	−52541.2	−2.10530
$855$	0	0
$856$	58.6515	0.00234190
$857$	−15421.4	−0.614687	−0.307343	−	0.951599i	$-0.599440\pi$
$857$	−15421.4	−0.614687	−0.307343	+	0.951599i	$0.599440\pi$
$858$	0	0
$859$	10191.5	0.404807	0.202403	−	0.979302i	$-0.435125\pi$
$859$	10191.5	0.404807	0.202403	+	0.979302i	$0.435125\pi$
$860$	−851.388	−0.0337583
$861$	0	0
$862$	−29690.1	−1.17314
$863$	−35480.0	−1.39948	−0.699742	−	0.714396i	$-0.746702\pi$
$863$	−35480.0	−1.39948	−0.699742	+	0.714396i	$0.746702\pi$
$864$	0	0
$865$	−14221.8	−0.559026
$866$	−5685.07	−0.223079
$867$	0	0
$868$	4666.67	0.182485
$869$	0	0
$870$	0	0
$871$	3467.00	0.134874
$872$	−229.656	−0.00891873
$873$	0	0
$874$	−3246.20	−0.125634
$875$	−42039.1	−1.62421
$876$	0	0
$877$	−18992.9	−0.731296	−0.365648	−	0.930753i	$-0.619153\pi$
$877$	−18992.9	−0.731296	−0.365648	+	0.930753i	$0.619153\pi$
$878$	−45014.3	−1.73025
$879$	0	0
$880$	0	0
$881$	43661.0	1.66967	0.834833	−	0.550504i	$-0.185564\pi$
$881$	43661.0	1.66967	0.834833	+	0.550504i	$0.185564\pi$
$882$	0	0
$883$	−20242.0	−0.771458	−0.385729	−	0.922612i	$-0.626050\pi$
$883$	−20242.0	−0.771458	−0.385729	+	0.922612i	$0.626050\pi$
$884$	146.030	0.00555600
$885$	0	0
$886$	38108.1	1.44500
$887$	20063.9	0.759503	0.379752	−	0.925088i	$-0.376009\pi$
$887$	20063.9	0.759503	0.379752	+	0.925088i	$0.376009\pi$
$888$	0	0
$889$	34951.6	1.31860
$890$	1057.02	0.0398106
$891$	0	0
$892$	1152.51	0.0432611
$893$	−5284.70	−0.198035
$894$	0	0
$895$	−20889.2	−0.780168
$896$	−50711.4	−1.89079
$897$	0	0
$898$	22753.3	0.845530
$899$	−10491.8	−0.389235
$900$	0	0
$901$	−2541.25	−0.0939637
$902$	0	0
$903$	0	0
$904$	49045.9	1.80447
$905$	22705.5	0.833983
$906$	0	0
$907$	−9772.55	−0.357764	−0.178882	−	0.983870i	$-0.557248\pi$
$907$	−9772.55	−0.357764	−0.178882	+	0.983870i	$0.557248\pi$
$908$	610.796	0.0223238
$909$	0	0
$910$	2608.83	0.0950350
$911$	9864.72	0.358763	0.179381	−	0.983780i	$-0.442590\pi$
$911$	9864.72	0.358763	0.179381	+	0.983780i	$0.442590\pi$
$912$	0	0
$913$	0	0
$914$	1478.70	0.0535132
$915$	0	0
$916$	465.420	0.0167881
$917$	9546.38	0.343783
$918$	0	0
$919$	−12906.0	−0.463253	−0.231627	−	0.972805i	$-0.574405\pi$
$919$	−12906.0	−0.463253	−0.231627	+	0.972805i	$0.574405\pi$
$920$	5155.05	0.184736
$921$	0	0
$922$	29154.9	1.04140
$923$	119.372	0.00425696
$924$	0	0
$925$	−9747.18	−0.346471
$926$	16195.3	0.574740
$927$	0	0
$928$	−972.656	−0.0344062
$929$	−51082.5	−1.80405	−0.902025	−	0.431684i	$-0.857920\pi$
$929$	−51082.5	−1.80405	−0.902025	+	0.431684i	$0.857920\pi$
$930$	0	0
$931$	−19980.4	−0.703365
$932$	−1724.61	−0.0606132
$933$	0	0
$934$	45740.0	1.60242
$935$	0	0
$936$	0	0
$937$	−10712.2	−0.373482	−0.186741	−	0.982409i	$-0.559792\pi$
$937$	−10712.2	−0.373482	−0.186741	+	0.982409i	$0.559792\pi$
$938$	−72405.1	−2.52037
$939$	0	0
$940$	−614.676	−0.0213282
$941$	46943.3	1.62626	0.813128	−	0.582085i	$-0.197763\pi$
$941$	46943.3	1.62626	0.813128	+	0.582085i	$0.197763\pi$
$942$	0	0
$943$	−5140.80	−0.177526
$944$	−12295.1	−0.423912
$945$	0	0
$946$	0	0
$947$	−8917.62	−0.306002	−0.153001	−	0.988226i	$-0.548894\pi$
$947$	−8917.62	−0.306002	−0.153001	+	0.988226i	$0.548894\pi$
$948$	0	0
$949$	−1673.63	−0.0572481
$950$	7360.93	0.251390
$951$	0	0
$952$	41637.7	1.41753
$953$	44396.9	1.50908	0.754541	−	0.656253i	$-0.227859\pi$
$953$	44396.9	1.50908	0.754541	+	0.656253i	$0.227859\pi$
$954$	0	0
$955$	−10986.3	−0.372259
$956$	2425.63	0.0820612
$957$	0	0
$958$	46247.3	1.55969
$959$	83243.0	2.80298
$960$	0	0
$961$	40995.5	1.37610
$962$	1475.49	0.0494507
$963$	0	0
$964$	1039.02	0.0347143
$965$	−18692.8	−0.623566
$966$	0	0
$967$	−21049.8	−0.700016	−0.350008	−	0.936747i	$-0.613821\pi$
$967$	−21049.8	−0.700016	−0.350008	+	0.936747i	$0.613821\pi$
$968$	0	0
$969$	0	0
$970$	10791.9	0.357223
$971$	5436.91	0.179690	0.0898450	−	0.995956i	$-0.471363\pi$
$971$	5436.91	0.179690	0.0898450	+	0.995956i	$0.471363\pi$
$972$	0	0
$973$	7825.09	0.257822
$974$	−3027.01	−0.0995809
$975$	0	0
$976$	−38080.4	−1.24890
$977$	−34449.4	−1.12808	−0.564040	−	0.825747i	$-0.690754\pi$
$977$	−34449.4	−1.12808	−0.564040	+	0.825747i	$0.690754\pi$
$978$	0	0
$979$	0	0
$980$	−2323.97	−0.0757516
$981$	0	0
$982$	27438.1	0.891633
$983$	16668.6	0.540840	0.270420	−	0.962742i	$-0.412837\pi$
$983$	16668.6	0.540840	0.270420	+	0.962742i	$0.412837\pi$
$984$	0	0
$985$	−27275.7	−0.882311
$986$	6856.47	0.221455
$987$	0	0
$988$	−71.1850	−0.00229220
$989$	9669.72	0.310899
$990$	0	0
$991$	26999.0	0.865441	0.432721	−	0.901528i	$-0.357554\pi$
$991$	26999.0	0.865441	0.432721	+	0.901528i	$0.357554\pi$
$992$	6562.33	0.210034
$993$	0	0
$994$	−2492.97	−0.0795496
$995$	25356.2	0.807886
$996$	0	0
$997$	19974.4	0.634499	0.317249	−	0.948342i	$-0.397241\pi$
$997$	19974.4	0.634499	0.317249	+	0.948342i	$0.397241\pi$
$998$	−13565.9	−0.430283
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.4.a.bh.1.3		4
3.2	odd	2		121.4.a.f.1.2		4
11.7	odd	10		99.4.f.c.82.1		8
11.8	odd	10		99.4.f.c.64.1		8
11.10	odd	2		1089.4.a.y.1.2		4
12.11	even	2		1936.4.a.bl.1.2		4
33.2	even	10		121.4.c.h.81.1		8
33.5	odd	10		121.4.c.b.3.2		8
33.8	even	10		11.4.c.a.9.2	yes	8
33.14	odd	10		121.4.c.i.9.1		8
33.17	even	10		121.4.c.h.3.1		8
33.20	odd	10		121.4.c.b.81.2		8
33.26	odd	10		121.4.c.i.27.1		8
33.29	even	10		11.4.c.a.5.2	✓	8
33.32	even	2		121.4.a.g.1.3		4
132.95	odd	10		176.4.m.c.49.2		8
132.107	odd	10		176.4.m.c.97.2		8
132.131	odd	2		1936.4.a.bk.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.c.a.5.2	✓	8	33.29	even	10
11.4.c.a.9.2	yes	8	33.8	even	10
99.4.f.c.64.1		8	11.8	odd	10
99.4.f.c.82.1		8	11.7	odd	10
121.4.a.f.1.2		4	3.2	odd	2
121.4.a.g.1.3		4	33.32	even	2
121.4.c.b.3.2		8	33.5	odd	10
121.4.c.b.81.2		8	33.20	odd	10
121.4.c.h.3.1		8	33.17	even	10
121.4.c.h.81.1		8	33.2	even	10
121.4.c.i.9.1		8	33.14	odd	10
121.4.c.i.27.1		8	33.26	odd	10
176.4.m.c.49.2		8	132.95	odd	10
176.4.m.c.97.2		8	132.107	odd	10
1089.4.a.y.1.2		4	11.10	odd	2
1089.4.a.bh.1.3		4	1.1	even	1	trivial
1936.4.a.bk.1.2		4	132.131	odd	2
1936.4.a.bl.1.2		4	12.11	even	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q+2.92335 q^{2} +0.545959 q^{4} -6.17671 q^{5} -32.1271 q^{7} -21.7907 q^{8} +O(q^{10})\)	\(q+2.92335 q^{2} +0.545959 q^{4} -6.17671 q^{5} -32.1271 q^{7} -21.7907 q^{8} -18.0567 q^{10} +4.49714 q^{13} -93.9186 q^{14} -68.0696 q^{16} +59.4763 q^{17} -28.9929 q^{19} -3.37223 q^{20} +38.3004 q^{23} -86.8483 q^{25} +13.1467 q^{26} -17.5401 q^{28} +39.4345 q^{29} -266.057 q^{31} -24.6651 q^{32} +173.870 q^{34} +198.440 q^{35} +112.232 q^{37} -84.7563 q^{38} +134.595 q^{40} -134.223 q^{41} +252.470 q^{43} +111.965 q^{46} +182.276 q^{47} +689.150 q^{49} -253.888 q^{50} +2.45526 q^{52} -42.7271 q^{53} +700.073 q^{56} +115.281 q^{58} +180.626 q^{59} +559.433 q^{61} -777.778 q^{62} +472.452 q^{64} -27.7775 q^{65} +770.935 q^{67} +32.4716 q^{68} +580.108 q^{70} +26.5440 q^{71} -372.155 q^{73} +328.094 q^{74} -15.8289 q^{76} +252.776 q^{79} +420.446 q^{80} -392.380 q^{82} +1055.18 q^{83} -367.368 q^{85} +738.058 q^{86} -58.5392 q^{89} -144.480 q^{91} +20.9105 q^{92} +532.855 q^{94} +179.081 q^{95} -597.668 q^{97} +2014.62 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 14 q^{4} + 11 q^{5} - 25 q^{7} + 66 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 14 * q^4 + 11 * q^5 - 25 * q^7 + 66 * q^8	\( 4 q + 4 q^{2} + 14 q^{4} + 11 q^{5} - 25 q^{7} + 66 q^{8} - 20 q^{10} - 25 q^{13} - 132 q^{14} + 34 q^{16} + 232 q^{17} - 154 q^{19} + 254 q^{20} + 6 q^{23} - 13 q^{25} + 200 q^{26} - 24 q^{28} + 363 q^{29} + 37 q^{31} + 162 q^{32} - 149 q^{34} + 356 q^{35} + 93 q^{37} - 379 q^{38} + 814 q^{40} + 152 q^{41} + 325 q^{43} + 1258 q^{46} + 869 q^{47} - 245 q^{49} - 1016 q^{50} + 1010 q^{52} - 811 q^{53} + 780 q^{56} + 956 q^{58} - 178 q^{59} + 105 q^{61} - 342 q^{62} + 818 q^{64} + 895 q^{65} + 43 q^{67} - 135 q^{68} - 22 q^{70} - 629 q^{71} + 270 q^{73} + 202 q^{74} + 121 q^{76} - 977 q^{79} - 122 q^{80} + 789 q^{82} + 1686 q^{83} + 721 q^{85} + 277 q^{86} - 1891 q^{89} - 80 q^{91} + 3450 q^{92} - 756 q^{94} + 1804 q^{95} - 1772 q^{97} + 1370 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 14 * q^4 + 11 * q^5 - 25 * q^7 + 66 * q^8 - 20 * q^10 - 25 * q^13 - 132 * q^14 + 34 * q^16 + 232 * q^17 - 154 * q^19 + 254 * q^20 + 6 * q^23 - 13 * q^25 + 200 * q^26 - 24 * q^28 + 363 * q^29 + 37 * q^31 + 162 * q^32 - 149 * q^34 + 356 * q^35 + 93 * q^37 - 379 * q^38 + 814 * q^40 + 152 * q^41 + 325 * q^43 + 1258 * q^46 + 869 * q^47 - 245 * q^49 - 1016 * q^50 + 1010 * q^52 - 811 * q^53 + 780 * q^56 + 956 * q^58 - 178 * q^59 + 105 * q^61 - 342 * q^62 + 818 * q^64 + 895 * q^65 + 43 * q^67 - 135 * q^68 - 22 * q^70 - 629 * q^71 + 270 * q^73 + 202 * q^74 + 121 * q^76 - 977 * q^79 - 122 * q^80 + 789 * q^82 + 1686 * q^83 + 721 * q^85 + 277 * q^86 - 1891 * q^89 - 80 * q^91 + 3450 * q^92 - 756 * q^94 + 1804 * q^95 - 1772 * q^97 + 1370 * q^98

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