LMFDB - Embedded newform 1089.3.c.m.604.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 77x^12 + 88x^11 - 577x^10 + 578x^9 + 1520x^8 + 1868x^7 - 1619x^6 - 16804x^5 + 32427x^4 + 43316x^3 - 71672x^2 + 83521
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.16
Root			$1.64608 + 1.06057i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.m.604.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.65113i q^{2} -9.33074 q^{4} -5.47004 q^{5} -7.16054i q^{7} -19.4632i q^{8} +O(q^{10})$	$q+3.65113i q^{2} -9.33074 q^{4} -5.47004 q^{5} -7.16054i q^{7} -19.4632i q^{8} -19.9718i q^{10} -6.59903i q^{13} +26.1440 q^{14} +33.7398 q^{16} +18.4591i q^{17} -8.42888i q^{19} +51.0395 q^{20} +17.7114 q^{23} +4.92132 q^{25} +24.0939 q^{26} +66.8132i q^{28} -20.0783i q^{29} +35.3426 q^{31} +45.3355i q^{32} -67.3966 q^{34} +39.1684i q^{35} -48.8236 q^{37} +30.7749 q^{38} +106.465i q^{40} +31.6316i q^{41} +45.0047i q^{43} +64.6665i q^{46} +0.728269 q^{47} -2.27330 q^{49} +17.9684i q^{50} +61.5739i q^{52} -70.1790 q^{53} -139.367 q^{56} +73.3084 q^{58} -24.1063 q^{59} +41.0918i q^{61} +129.040i q^{62} -30.5665 q^{64} +36.0970i q^{65} +96.0426 q^{67} -172.237i q^{68} -143.009 q^{70} -39.5948 q^{71} +70.2593i q^{73} -178.261i q^{74} +78.6477i q^{76} -89.3597i q^{79} -184.558 q^{80} -115.491 q^{82} +26.2209i q^{83} -100.972i q^{85} -164.318 q^{86} +118.861 q^{89} -47.2526 q^{91} -165.260 q^{92} +2.65900i q^{94} +46.1063i q^{95} -33.1037 q^{97} -8.30011i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) $ 16 * q - 20 * q^4 + 4 * q^5	$ 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) $ 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$1$

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 $

$2$	3.65113i	1.82556i	0.408446	+	0.912782i	$0.366071\pi$
$2$	3.65113i	1.82556i	−0.408446	+	0.912782i	$0.633929\pi$
$3$	0	0
$3$	0	0
$4$	−9.33074	−2.33269
$5$	−5.47004	−1.09401	−0.547004	−	0.837130i	$-0.684232\pi$
$5$	−5.47004	−1.09401	−0.547004	+	0.837130i	$0.684232\pi$
$6$	0	0
$7$	− 7.16054i	− 1.02293i	−0.859303	−	0.511467i	$-0.829102\pi$
$7$	− 7.16054i	− 1.02293i	0.859303	−	0.511467i	$-0.170898\pi$
$8$	− 19.4632i	− 2.43290i
$9$	0	0
$10$	− 19.9718i	− 1.99718i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 6.59903i	− 0.507618i	−0.967254	−	0.253809i	$-0.918317\pi$
$13$	− 6.59903i	− 0.507618i	0.967254	−	0.253809i	$-0.0816834\pi$
$14$	26.1440	1.86743
$15$	0	0
$16$	33.7398	2.10874
$17$	18.4591i	1.08583i	0.839788	+	0.542915i	$0.182679\pi$
$17$	18.4591i	1.08583i	−0.839788	+	0.542915i	$0.817321\pi$
$18$	0	0
$19$	− 8.42888i	− 0.443625i	−0.975089	−	0.221813i	$-0.928803\pi$
$19$	− 8.42888i	− 0.443625i	0.975089	−	0.221813i	$-0.0711973\pi$
$20$	51.0395	2.55198
$21$	0	0
$22$	0	0
$23$	17.7114	0.770060	0.385030	−	0.922904i	$-0.374191\pi$
$23$	17.7114	0.770060	0.385030	+	0.922904i	$0.374191\pi$
$24$	0	0
$25$	4.92132	0.196853
$26$	24.0939	0.926689
$27$	0	0
$28$	66.8132i	2.38618i
$29$	− 20.0783i	− 0.692354i	−0.938169	−	0.346177i	$-0.887480\pi$
$29$	− 20.0783i	− 0.692354i	0.938169	−	0.346177i	$-0.112520\pi$
$30$	0	0
$31$	35.3426	1.14008	0.570042	−	0.821615i	$-0.306927\pi$
$31$	35.3426	1.14008	0.570042	+	0.821615i	$0.306927\pi$
$32$	45.3355i	1.41673i
$33$	0	0
$34$	−67.3966	−1.98225
$35$	39.1684i	1.11910i
$36$	0	0
$37$	−48.8236	−1.31956	−0.659778	−	0.751460i	$-0.729350\pi$
$37$	−48.8236	−1.31956	−0.659778	+	0.751460i	$0.729350\pi$
$38$	30.7749	0.809866
$39$	0	0
$40$	106.465i	2.66162i
$41$	31.6316i	0.771504i	0.922603	+	0.385752i	$0.126058\pi$
$41$	31.6316i	0.771504i	−0.922603	+	0.385752i	$0.873942\pi$
$42$	0	0
$43$	45.0047i	1.04662i	0.852142	+	0.523311i	$0.175303\pi$
$43$	45.0047i	1.04662i	−0.852142	+	0.523311i	$0.824697\pi$
$44$	0	0
$45$	0	0
$46$	64.6665i	1.40579i
$47$	0.728269	0.0154951	0.00774754	−	0.999970i	$-0.497534\pi$
$47$	0.728269	0.0154951	0.00774754	+	0.999970i	$0.497534\pi$
$48$	0	0
$49$	−2.27330	−0.0463939
$50$	17.9684i	0.359368i
$51$	0	0
$52$	61.5739i	1.18411i
$53$	−70.1790	−1.32413	−0.662066	−	0.749446i	$-0.730320\pi$
$53$	−70.1790	−1.32413	−0.662066	+	0.749446i	$0.730320\pi$
$54$	0	0
$55$	0	0
$56$	−139.367	−2.48870
$57$	0	0
$58$	73.3084	1.26394
$59$	−24.1063	−0.408582	−0.204291	−	0.978910i	$-0.565489\pi$
$59$	−24.1063	−0.408582	−0.204291	+	0.978910i	$0.565489\pi$
$60$	0	0
$61$	41.0918i	0.673636i	0.941570	+	0.336818i	$0.109351\pi$
$61$	41.0918i	0.673636i	−0.941570	+	0.336818i	$0.890649\pi$
$62$	129.040i	2.08130i
$63$	0	0
$64$	−30.5665	−0.477601
$65$	36.0970i	0.555338i
$66$	0	0
$67$	96.0426	1.43347	0.716736	−	0.697345i	$-0.245635\pi$
$67$	96.0426	1.43347	0.716736	+	0.697345i	$0.245635\pi$
$68$	− 172.237i	− 2.53290i
$69$	0	0
$70$	−143.009	−2.04299
$71$	−39.5948	−0.557674	−0.278837	−	0.960338i	$-0.589949\pi$
$71$	−39.5948	−0.557674	−0.278837	+	0.960338i	$0.589949\pi$
$72$	0	0
$73$	70.2593i	0.962456i	0.876595	+	0.481228i	$0.159809\pi$
$73$	70.2593i	0.962456i	−0.876595	+	0.481228i	$0.840191\pi$
$74$	− 178.261i	− 2.40894i
$75$	0	0
$76$	78.6477i	1.03484i
$77$	0	0
$78$	0	0
$79$	− 89.3597i	− 1.13114i	−0.824702	−	0.565568i	$-0.808657\pi$
$79$	− 89.3597i	− 1.13114i	0.824702	−	0.565568i	$-0.191343\pi$
$80$	−184.558	−2.30698
$81$	0	0
$82$	−115.491	−1.40843
$83$	26.2209i	0.315914i	0.987446	+	0.157957i	$0.0504907\pi$
$83$	26.2209i	0.315914i	−0.987446	+	0.157957i	$0.949509\pi$
$84$	0	0
$85$	− 100.972i	− 1.18791i
$86$	−164.318	−1.91067
$87$	0	0
$88$	0	0
$89$	118.861	1.33552	0.667760	−	0.744377i	$-0.267253\pi$
$89$	118.861	1.33552	0.667760	+	0.744377i	$0.267253\pi$
$90$	0	0
$91$	−47.2526	−0.519259
$92$	−165.260	−1.79631
$93$	0	0
$94$	2.65900i	0.0282873i
$95$	46.1063i	0.485329i
$96$	0	0
$97$	−33.1037	−0.341275	−0.170637	−	0.985334i	$-0.554583\pi$
$97$	−33.1037	−0.341275	−0.170637	+	0.985334i	$0.554583\pi$
$98$	− 8.30011i	− 0.0846950i
$99$	0	0
$100$	−45.9196	−0.459196
$101$	− 51.3115i	− 0.508035i	−0.967200	−	0.254017i	$-0.918248\pi$
$101$	− 51.3115i	− 0.508035i	0.967200	−	0.254017i	$-0.0817521\pi$
$102$	0	0
$103$	121.005	1.17480	0.587402	−	0.809295i	$-0.300151\pi$
$103$	121.005	1.17480	0.587402	+	0.809295i	$0.300151\pi$
$104$	−128.439	−1.23499
$105$	0	0
$106$	− 256.232i	− 2.41729i
$107$	177.477i	1.65866i	0.558759	+	0.829330i	$0.311277\pi$
$107$	177.477i	1.65866i	−0.558759	+	0.829330i	$0.688723\pi$
$108$	0	0
$109$	81.6242i	0.748846i	0.927258	+	0.374423i	$0.122159\pi$
$109$	81.6242i	0.748846i	−0.927258	+	0.374423i	$0.877841\pi$
$110$	0	0
$111$	0	0
$112$	− 241.595i	− 2.15710i
$113$	21.3922	0.189311	0.0946555	−	0.995510i	$-0.469825\pi$
$113$	21.3922	0.189311	0.0946555	+	0.995510i	$0.469825\pi$
$114$	0	0
$115$	−96.8819	−0.842452
$116$	187.345i	1.61504i
$117$	0	0
$118$	− 88.0153i	− 0.745892i
$119$	132.177	1.11073
$120$	0	0
$121$	0	0
$122$	−150.031	−1.22977
$123$	0	0
$124$	−329.773	−2.65946
$125$	109.831	0.878649
$126$	0	0
$127$	− 77.9657i	− 0.613903i	−0.951725	−	0.306952i	$-0.900691\pi$
$127$	− 77.9657i	− 0.613903i	0.951725	−	0.306952i	$-0.0993090\pi$
$128$	69.7399i	0.544843i
$129$	0	0
$130$	−131.795	−1.01380
$131$	74.9602i	0.572215i	0.958197	+	0.286108i	$0.0923615\pi$
$131$	74.9602i	0.572215i	−0.958197	+	0.286108i	$0.907638\pi$
$132$	0	0
$133$	−60.3553	−0.453799
$134$	350.664i	2.61690i
$135$	0	0
$136$	359.274	2.64172
$137$	−158.426	−1.15639	−0.578197	−	0.815897i	$-0.696244\pi$
$137$	−158.426	−1.15639	−0.578197	+	0.815897i	$0.696244\pi$
$138$	0	0
$139$	258.173i	1.85736i	0.370885	+	0.928679i	$0.379054\pi$
$139$	258.173i	1.85736i	−0.370885	+	0.928679i	$0.620946\pi$
$140$	− 365.471i	− 2.61050i
$141$	0	0
$142$	− 144.566i	− 1.01807i
$143$	0	0
$144$	0	0
$145$	109.829i	0.757441i
$146$	−256.526	−1.75703
$147$	0	0
$148$	455.560	3.07811
$149$	56.2236i	0.377340i	0.982041	+	0.188670i	$0.0604176\pi$
$149$	56.2236i	0.377340i	−0.982041	+	0.188670i	$0.939582\pi$
$150$	0	0
$151$	113.602i	0.752334i	0.926552	+	0.376167i	$0.122758\pi$
$151$	113.602i	0.752334i	−0.926552	+	0.376167i	$0.877242\pi$
$152$	−164.053	−1.07930
$153$	0	0
$154$	0	0
$155$	−193.325	−1.24726
$156$	0	0
$157$	267.088	1.70120	0.850599	−	0.525816i	$-0.176240\pi$
$157$	267.088	1.70120	0.850599	+	0.525816i	$0.176240\pi$
$158$	326.264	2.06496
$159$	0	0
$160$	− 247.987i	− 1.54992i
$161$	− 126.823i	− 0.787721i
$162$	0	0
$163$	240.252	1.47394	0.736971	−	0.675925i	$-0.236256\pi$
$163$	240.252	1.47394	0.736971	+	0.675925i	$0.236256\pi$
$164$	− 295.147i	− 1.79968i
$165$	0	0
$166$	−95.7357	−0.576721
$167$	176.322i	1.05582i	0.849301	+	0.527910i	$0.177024\pi$
$167$	176.322i	1.05582i	−0.849301	+	0.527910i	$0.822976\pi$
$168$	0	0
$169$	125.453	0.742324
$170$	368.662	2.16860
$171$	0	0
$172$	− 419.927i	− 2.44144i
$173$	− 266.798i	− 1.54219i	−0.636723	−	0.771093i	$-0.719710\pi$
$173$	− 266.798i	− 1.54219i	0.636723	−	0.771093i	$-0.280290\pi$
$174$	0	0
$175$	− 35.2393i	− 0.201367i
$176$	0	0
$177$	0	0
$178$	433.978i	2.43808i
$179$	57.9328	0.323647	0.161823	−	0.986820i	$-0.448262\pi$
$179$	57.9328	0.323647	0.161823	+	0.986820i	$0.448262\pi$
$180$	0	0
$181$	−232.528	−1.28469	−0.642343	−	0.766417i	$-0.722038\pi$
$181$	−232.528	−1.28469	−0.642343	+	0.766417i	$0.722038\pi$
$182$	− 172.525i	− 0.947942i
$183$	0	0
$184$	− 344.721i	− 1.87348i
$185$	267.067	1.44360
$186$	0	0
$187$	0	0
$188$	−6.79529	−0.0361452
$189$	0	0
$190$	−168.340	−0.886000
$191$	−13.7011	−0.0717335	−0.0358667	−	0.999357i	$-0.511419\pi$
$191$	−13.7011	−0.0717335	−0.0358667	+	0.999357i	$0.511419\pi$
$192$	0	0
$193$	34.5413i	0.178971i	0.995988	+	0.0894853i	$0.0285222\pi$
$193$	34.5413i	0.178971i	−0.995988	+	0.0894853i	$0.971478\pi$
$194$	− 120.866i	− 0.623019i
$195$	0	0
$196$	21.2116	0.108222
$197$	103.908i	0.527451i	0.964598	+	0.263726i	$0.0849514\pi$
$197$	103.908i	0.527451i	−0.964598	+	0.263726i	$0.915049\pi$
$198$	0	0
$199$	264.816	1.33073	0.665367	−	0.746516i	$-0.268275\pi$
$199$	264.816	1.33073	0.665367	+	0.746516i	$0.268275\pi$
$200$	− 95.7849i	− 0.478924i
$201$	0	0
$202$	187.345	0.927450
$203$	−143.771	−0.708232
$204$	0	0
$205$	− 173.026i	− 0.844031i
$206$	441.804i	2.14468i
$207$	0	0
$208$	− 222.650i	− 1.07043i
$209$	0	0
$210$	0	0
$211$	294.987i	1.39804i	0.715100	+	0.699022i	$0.246381\pi$
$211$	294.987i	1.39804i	−0.715100	+	0.699022i	$0.753619\pi$
$212$	654.822	3.08878
$213$	0	0
$214$	−647.990	−3.02799
$215$	− 246.177i	− 1.14501i
$216$	0	0
$217$	− 253.072i	− 1.16623i
$218$	−298.020	−1.36707
$219$	0	0
$220$	0	0
$221$	121.812	0.551186
$222$	0	0
$223$	206.820	0.927442	0.463721	−	0.885981i	$-0.346514\pi$
$223$	206.820	0.927442	0.463721	+	0.885981i	$0.346514\pi$
$224$	324.627	1.44923
$225$	0	0
$226$	78.1055i	0.345600i
$227$	− 288.893i	− 1.27266i	−0.771418	−	0.636328i	$-0.780452\pi$
$227$	− 288.893i	− 1.27266i	0.771418	−	0.636328i	$-0.219548\pi$
$228$	0	0
$229$	−385.408	−1.68300	−0.841502	−	0.540254i	$-0.818328\pi$
$229$	−385.408	−1.68300	−0.841502	+	0.540254i	$0.818328\pi$
$230$	− 353.728i	− 1.53795i
$231$	0	0
$232$	−390.788	−1.68443
$233$	− 141.874i	− 0.608900i	−0.952528	−	0.304450i	$-0.901527\pi$
$233$	− 141.874i	− 0.608900i	0.952528	−	0.304450i	$-0.0984727\pi$
$234$	0	0
$235$	−3.98366	−0.0169517
$236$	224.930	0.953093
$237$	0	0
$238$	482.596i	2.02771i
$239$	− 200.660i	− 0.839581i	−0.907621	−	0.419791i	$-0.862103\pi$
$239$	− 200.660i	− 0.839581i	0.907621	−	0.419791i	$-0.137897\pi$
$240$	0	0
$241$	153.553i	0.637150i	0.947898	+	0.318575i	$0.103204\pi$
$241$	153.553i	0.637150i	−0.947898	+	0.318575i	$0.896796\pi$
$242$	0	0
$243$	0	0
$244$	− 383.417i	− 1.57138i
$245$	12.4350	0.0507552
$246$	0	0
$247$	−55.6224	−0.225192
$248$	− 687.882i	− 2.77372i
$249$	0	0
$250$	401.008i	1.60403i
$251$	183.445	0.730856	0.365428	−	0.930840i	$-0.380923\pi$
$251$	183.445	0.730856	0.365428	+	0.930840i	$0.380923\pi$
$252$	0	0
$253$	0	0
$254$	284.663	1.12072
$255$	0	0
$256$	−376.895	−1.47225
$257$	−199.127	−0.774812	−0.387406	−	0.921909i	$-0.626629\pi$
$257$	−199.127	−0.774812	−0.387406	+	0.921909i	$0.626629\pi$
$258$	0	0
$259$	349.603i	1.34982i
$260$	− 336.811i	− 1.29543i
$261$	0	0
$262$	−273.689	−1.04462
$263$	241.980i	0.920077i	0.887899	+	0.460039i	$0.152164\pi$
$263$	241.980i	0.920077i	−0.887899	+	0.460039i	$0.847836\pi$
$264$	0	0
$265$	383.882	1.44861
$266$	− 220.365i	− 0.828440i
$267$	0	0
$268$	−896.149	−3.34384
$269$	11.3277	0.0421106	0.0210553	−	0.999778i	$-0.493297\pi$
$269$	11.3277	0.0421106	0.0210553	+	0.999778i	$0.493297\pi$
$270$	0	0
$271$	− 229.609i	− 0.847264i	−0.905834	−	0.423632i	$-0.860755\pi$
$271$	− 229.609i	− 0.847264i	0.905834	−	0.423632i	$-0.139245\pi$
$272$	622.807i	2.28973i
$273$	0	0
$274$	− 578.434i	− 2.11107i
$275$	0	0
$276$	0	0
$277$	331.466i	1.19663i	0.801262	+	0.598313i	$0.204162\pi$
$277$	331.466i	1.19663i	−0.801262	+	0.598313i	$0.795838\pi$
$278$	−942.622	−3.39073
$279$	0	0
$280$	762.344	2.72266
$281$	399.606i	1.42209i	0.703149	+	0.711043i	$0.251777\pi$
$281$	399.606i	1.42209i	−0.703149	+	0.711043i	$0.748223\pi$
$282$	0	0
$283$	93.0858i	0.328925i	0.986383	+	0.164462i	$0.0525889\pi$
$283$	93.0858i	0.328925i	−0.986383	+	0.164462i	$0.947411\pi$
$284$	369.449	1.30088
$285$	0	0
$286$	0	0
$287$	226.500	0.789197
$288$	0	0
$289$	−51.7386	−0.179026
$290$	−401.000	−1.38276
$291$	0	0
$292$	− 655.572i	− 2.24511i
$293$	245.937i	0.839377i	0.907668	+	0.419688i	$0.137861\pi$
$293$	245.937i	0.839377i	−0.907668	+	0.419688i	$0.862139\pi$
$294$	0	0
$295$	131.862	0.446991
$296$	950.265i	3.21036i
$297$	0	0
$298$	−205.280	−0.688858
$299$	− 116.878i	− 0.390896i
$300$	0	0
$301$	322.258	1.07062
$302$	−414.777	−1.37343
$303$	0	0
$304$	− 284.389i	− 0.935489i
$305$	− 224.774i	− 0.736963i
$306$	0	0
$307$	− 2.17423i	− 0.00708218i	−0.999994	−	0.00354109i	$-0.998873\pi$
$307$	− 2.17423i	− 0.00708218i	0.999994	−	0.00354109i	$-0.00112717\pi$
$308$	0	0
$309$	0	0
$310$	− 705.856i	− 2.27696i
$311$	317.733	1.02165	0.510824	−	0.859685i	$-0.329340\pi$
$311$	317.733	1.02165	0.510824	+	0.859685i	$0.329340\pi$
$312$	0	0
$313$	−100.641	−0.321536	−0.160768	−	0.986992i	$-0.551397\pi$
$313$	−100.641	−0.321536	−0.160768	+	0.986992i	$0.551397\pi$
$314$	975.173i	3.10565i
$315$	0	0
$316$	833.792i	2.63858i
$317$	−279.045	−0.880267	−0.440133	−	0.897932i	$-0.645069\pi$
$317$	−279.045	−0.880267	−0.440133	+	0.897932i	$0.645069\pi$
$318$	0	0
$319$	0	0
$320$	167.200	0.522499
$321$	0	0
$322$	463.047	1.43803
$323$	155.590	0.481701
$324$	0	0
$325$	− 32.4760i	− 0.0999260i
$326$	877.193i	2.69078i
$327$	0	0
$328$	615.654	1.87699
$329$	− 5.21480i	− 0.0158504i
$330$	0	0
$331$	−64.8831	−0.196021	−0.0980107	−	0.995185i	$-0.531248\pi$
$331$	−64.8831	−0.196021	−0.0980107	+	0.995185i	$0.531248\pi$
$332$	− 244.660i	− 0.736928i
$333$	0	0
$334$	−643.774	−1.92747
$335$	−525.357	−1.56823
$336$	0	0
$337$	557.406i	1.65402i	0.562184	+	0.827012i	$0.309961\pi$
$337$	557.406i	1.65402i	−0.562184	+	0.827012i	$0.690039\pi$
$338$	458.044i	1.35516i
$339$	0	0
$340$	942.144i	2.77101i
$341$	0	0
$342$	0	0
$343$	− 334.588i	− 0.975476i
$344$	875.937	2.54633
$345$	0	0
$346$	974.114	2.81536
$347$	− 655.452i	− 1.88891i	−0.328641	−	0.944455i	$-0.606591\pi$
$347$	− 655.452i	− 1.88891i	0.328641	−	0.944455i	$-0.393409\pi$
$348$	0	0
$349$	16.2405i	0.0465343i	0.999729	+	0.0232672i	$0.00740683\pi$
$349$	16.2405i	0.0465343i	−0.999729	+	0.0232672i	$0.992593\pi$
$350$	128.663	0.367609
$351$	0	0
$352$	0	0
$353$	372.243	1.05451	0.527256	−	0.849706i	$-0.323221\pi$
$353$	372.243	1.05451	0.527256	+	0.849706i	$0.323221\pi$
$354$	0	0
$355$	216.585	0.610099
$356$	−1109.06	−3.11535
$357$	0	0
$358$	211.520i	0.590838i
$359$	− 57.9743i	− 0.161488i	−0.996735	−	0.0807442i	$-0.974270\pi$
$359$	− 57.9743i	− 0.161488i	0.996735	−	0.0807442i	$-0.0257297\pi$
$360$	0	0
$361$	289.954	0.803197
$362$	− 848.991i	− 2.34528i
$363$	0	0
$364$	440.902	1.21127
$365$	− 384.321i	− 1.05293i
$366$	0	0
$367$	−188.083	−0.512487	−0.256244	−	0.966612i	$-0.582485\pi$
$367$	−188.083	−0.512487	−0.256244	+	0.966612i	$0.582485\pi$
$368$	597.579	1.62386
$369$	0	0
$370$	975.096i	2.63539i
$371$	502.519i	1.35450i
$372$	0	0
$373$	470.904i	1.26248i	0.775589	+	0.631238i	$0.217453\pi$
$373$	470.904i	1.26248i	−0.775589	+	0.631238i	$0.782547\pi$
$374$	0	0
$375$	0	0
$376$	− 14.1745i	− 0.0376981i
$377$	−132.497	−0.351451
$378$	0	0
$379$	282.963	0.746604	0.373302	−	0.927710i	$-0.378225\pi$
$379$	282.963	0.746604	0.373302	+	0.927710i	$0.378225\pi$
$380$	− 430.206i	− 1.13212i
$381$	0	0
$382$	− 50.0245i	− 0.130954i
$383$	−69.2026	−0.180686	−0.0903428	−	0.995911i	$-0.528796\pi$
$383$	−69.2026	−0.180686	−0.0903428	+	0.995911i	$0.528796\pi$
$384$	0	0
$385$	0	0
$386$	−126.115	−0.326722
$387$	0	0
$388$	308.882	0.796087
$389$	590.710	1.51854	0.759268	−	0.650778i	$-0.225557\pi$
$389$	590.710	1.51854	0.759268	+	0.650778i	$0.225557\pi$
$390$	0	0
$391$	326.936i	0.836154i
$392$	44.2458i	0.112872i
$393$	0	0
$394$	−379.381	−0.962897
$395$	488.801i	1.23747i
$396$	0	0
$397$	−492.120	−1.23960	−0.619798	−	0.784761i	$-0.712786\pi$
$397$	−492.120	−1.23960	−0.619798	+	0.784761i	$0.712786\pi$
$398$	966.878i	2.42934i
$399$	0	0
$400$	166.045	0.415111
$401$	544.074	1.35679	0.678396	−	0.734696i	$-0.262675\pi$
$401$	544.074	1.35679	0.678396	+	0.734696i	$0.262675\pi$
$402$	0	0
$403$	− 233.227i	− 0.578727i
$404$	478.775i	1.18509i
$405$	0	0
$406$	− 524.927i	− 1.29292i
$407$	0	0
$408$	0	0
$409$	− 267.354i	− 0.653678i	−0.945080	−	0.326839i	$-0.894017\pi$
$409$	− 267.354i	− 0.653678i	0.945080	−	0.326839i	$-0.105983\pi$
$410$	631.741	1.54083
$411$	0	0
$412$	−1129.07	−2.74045
$413$	172.614i	0.417952i
$414$	0	0
$415$	− 143.429i	− 0.345612i
$416$	299.170	0.719159
$417$	0	0
$418$	0	0
$419$	−4.31451	−0.0102972	−0.00514858	−	0.999987i	$-0.501639\pi$
$419$	−4.31451	−0.0102972	−0.00514858	+	0.999987i	$0.501639\pi$
$420$	0	0
$421$	−324.534	−0.770864	−0.385432	−	0.922736i	$-0.625948\pi$
$421$	−324.534	−0.770864	−0.385432	+	0.922736i	$0.625948\pi$
$422$	−1077.04	−2.55222
$423$	0	0
$424$	1365.91i	3.22149i
$425$	90.8432i	0.213749i
$426$	0	0
$427$	294.239	0.689085
$428$	− 1655.99i	− 3.86913i
$429$	0	0
$430$	898.826	2.09029
$431$	78.6920i	0.182580i	0.995824	+	0.0912900i	$0.0290990\pi$
$431$	78.6920i	0.182580i	−0.995824	+	0.0912900i	$0.970901\pi$
$432$	0	0
$433$	−345.858	−0.798749	−0.399374	−	0.916788i	$-0.630773\pi$
$433$	−345.858	−0.798749	−0.399374	+	0.916788i	$0.630773\pi$
$434$	923.999	2.12903
$435$	0	0
$436$	− 761.615i	− 1.74682i
$437$	− 149.287i	− 0.341618i
$438$	0	0
$439$	− 380.781i	− 0.867382i	−0.901062	−	0.433691i	$-0.857211\pi$
$439$	− 380.781i	− 0.867382i	0.901062	−	0.433691i	$-0.142789\pi$
$440$	0	0
$441$	0	0
$442$	444.752i	1.00623i
$443$	165.213	0.372941	0.186470	−	0.982461i	$-0.440295\pi$
$443$	165.213	0.372941	0.186470	+	0.982461i	$0.440295\pi$
$444$	0	0
$445$	−650.176	−1.46107
$446$	755.125i	1.69311i
$447$	0	0
$448$	218.872i	0.488554i
$449$	465.113	1.03589	0.517943	−	0.855415i	$-0.326698\pi$
$449$	465.113	1.03589	0.517943	+	0.855415i	$0.326698\pi$
$450$	0	0
$451$	0	0
$452$	−199.605	−0.441603
$453$	0	0
$454$	1054.79	2.32332
$455$	258.474	0.568074
$456$	0	0
$457$	− 708.027i	− 1.54929i	−0.632394	−	0.774647i	$-0.717928\pi$
$457$	− 708.027i	− 1.54929i	0.632394	−	0.774647i	$-0.282072\pi$
$458$	− 1407.17i	− 3.07243i
$459$	0	0
$460$	903.981	1.96518
$461$	− 149.958i	− 0.325288i	−0.986685	−	0.162644i	$-0.947998\pi$
$461$	− 149.958i	− 0.325288i	0.986685	−	0.162644i	$-0.0520021\pi$
$462$	0	0
$463$	170.995	0.369319	0.184659	−	0.982803i	$-0.440882\pi$
$463$	170.995	0.369319	0.184659	+	0.982803i	$0.440882\pi$
$464$	− 677.437i	− 1.45999i
$465$	0	0
$466$	518.000	1.11159
$467$	659.861	1.41298	0.706489	−	0.707724i	$-0.250278\pi$
$467$	659.861	1.41298	0.706489	+	0.707724i	$0.250278\pi$
$468$	0	0
$469$	− 687.717i	− 1.46635i
$470$	− 14.5449i	− 0.0309465i
$471$	0	0
$472$	469.187i	0.994040i
$473$	0	0
$474$	0	0
$475$	− 41.4812i	− 0.0873289i
$476$	−1233.31	−2.59099
$477$	0	0
$478$	732.635	1.53271
$479$	258.880i	0.540458i	0.962796	+	0.270229i	$0.0870995\pi$
$479$	258.880i	0.540458i	−0.962796	+	0.270229i	$0.912901\pi$
$480$	0	0
$481$	322.188i	0.669830i
$482$	−560.643	−1.16316
$483$	0	0
$484$	0	0
$485$	181.078	0.373357
$486$	0	0
$487$	−53.0151	−0.108861	−0.0544303	−	0.998518i	$-0.517334\pi$
$487$	−53.0151	−0.108861	−0.0544303	+	0.998518i	$0.517334\pi$
$488$	799.779	1.63889
$489$	0	0
$490$	45.4019i	0.0926570i
$491$	341.371i	0.695256i	0.937632	+	0.347628i	$0.113013\pi$
$491$	341.371i	0.695256i	−0.937632	+	0.347628i	$0.886987\pi$
$492$	0	0
$493$	370.627	0.751779
$494$	− 203.085i	− 0.411102i
$495$	0	0
$496$	1192.45	2.40414
$497$	283.520i	0.570463i
$498$	0	0
$499$	−175.368	−0.351439	−0.175720	−	0.984440i	$-0.556225\pi$
$499$	−175.368	−0.351439	−0.175720	+	0.984440i	$0.556225\pi$
$500$	−1024.81	−2.04961
$501$	0	0
$502$	669.781i	1.33423i
$503$	256.856i	0.510648i	0.966856	+	0.255324i	$0.0821821\pi$
$503$	256.856i	0.510648i	−0.966856	+	0.255324i	$0.917818\pi$
$504$	0	0
$505$	280.676i	0.555794i
$506$	0	0
$507$	0	0
$508$	727.478i	1.43204i
$509$	25.4033	0.0499082	0.0249541	−	0.999689i	$-0.492056\pi$
$509$	25.4033	0.0499082	0.0249541	+	0.999689i	$0.492056\pi$
$510$	0	0
$511$	503.094	0.984529
$512$	− 1097.13i	− 2.14284i
$513$	0	0
$514$	− 727.037i	− 1.41447i
$515$	−661.901	−1.28525
$516$	0	0
$517$	0	0
$518$	−1276.45	−2.46418
$519$	0	0
$520$	702.564	1.35108
$521$	838.798	1.60998	0.804988	−	0.593291i	$-0.202172\pi$
$521$	838.798	1.60998	0.804988	+	0.593291i	$0.202172\pi$
$522$	0	0
$523$	− 78.3483i	− 0.149806i	−0.997191	−	0.0749028i	$-0.976135\pi$
$523$	− 78.3483i	− 0.149806i	0.997191	−	0.0749028i	$-0.0238646\pi$
$524$	− 699.435i	− 1.33480i
$525$	0	0
$526$	−883.501	−1.67966
$527$	652.393i	1.23794i
$528$	0	0
$529$	−215.307	−0.407008
$530$	1401.60i	2.64453i
$531$	0	0
$532$	563.160	1.05857
$533$	208.738	0.391629
$534$	0	0
$535$	− 970.804i	− 1.81459i
$536$	− 1869.30i	− 3.48750i
$537$	0	0
$538$	41.3591i	0.0768756i
$539$	0	0
$540$	0	0
$541$	− 306.326i	− 0.566223i	−0.959087	−	0.283111i	$-0.908633\pi$
$541$	− 306.326i	− 0.566223i	0.959087	−	0.283111i	$-0.0913666\pi$
$542$	838.331	1.54674
$543$	0	0
$544$	−836.853	−1.53833
$545$	− 446.487i	− 0.819243i
$546$	0	0
$547$	70.4130i	0.128726i	0.997927	+	0.0643629i	$0.0205015\pi$
$547$	70.4130i	0.128726i	−0.997927	+	0.0643629i	$0.979498\pi$
$548$	1478.23	2.69751
$549$	0	0
$550$	0	0
$551$	−169.237	−0.307146
$552$	0	0
$553$	−639.863	−1.15708
$554$	−1210.22	−2.18452
$555$	0	0
$556$	− 2408.94i	− 4.33263i
$557$	575.149i	1.03258i	0.856413	+	0.516292i	$0.172688\pi$
$557$	575.149i	1.03258i	−0.856413	+	0.516292i	$0.827312\pi$
$558$	0	0
$559$	296.987	0.531283
$560$	1321.54i	2.35988i
$561$	0	0
$562$	−1459.01	−2.59611
$563$	368.620i	0.654742i	0.944896	+	0.327371i	$0.106163\pi$
$563$	368.620i	0.654742i	−0.944896	+	0.327371i	$0.893837\pi$
$564$	0	0
$565$	−117.016	−0.207108
$566$	−339.868	−0.600474
$567$	0	0
$568$	770.644i	1.35677i
$569$	− 275.879i	− 0.484848i	−0.970170	−	0.242424i	$-0.922057\pi$
$569$	− 275.879i	− 0.484848i	0.970170	−	0.242424i	$-0.0779425\pi$
$570$	0	0
$571$	57.9706i	0.101525i	0.998711	+	0.0507624i	$0.0161651\pi$
$571$	57.9706i	0.101525i	−0.998711	+	0.0507624i	$0.983835\pi$
$572$	0	0
$573$	0	0
$574$	826.979i	1.44073i
$575$	87.1634	0.151589
$576$	0	0
$577$	−196.282	−0.340177	−0.170089	−	0.985429i	$-0.554405\pi$
$577$	−196.282	−0.340177	−0.170089	+	0.985429i	$0.554405\pi$
$578$	− 188.904i	− 0.326824i
$579$	0	0
$580$	− 1024.79i	− 1.76687i
$581$	187.755	0.323159
$582$	0	0
$583$	0	0
$584$	1367.47	2.34156
$585$	0	0
$586$	−897.949	−1.53234
$587$	−713.720	−1.21588	−0.607939	−	0.793984i	$-0.708003\pi$
$587$	−713.720	−1.21588	−0.607939	+	0.793984i	$0.708003\pi$
$588$	0	0
$589$	− 297.899i	− 0.505770i
$590$	481.447i	0.816012i
$591$	0	0
$592$	−1647.30	−2.78260
$593$	− 279.318i	− 0.471025i	−0.971871	−	0.235513i	$-0.924323\pi$
$593$	− 279.318i	− 0.471025i	0.971871	−	0.235513i	$-0.0756769\pi$
$594$	0	0
$595$	−723.014	−1.21515
$596$	− 524.608i	− 0.880215i
$597$	0	0
$598$	426.736	0.713606
$599$	−660.906	−1.10335	−0.551674	−	0.834060i	$-0.686011\pi$
$599$	−660.906	−1.10335	−0.551674	+	0.834060i	$0.686011\pi$
$600$	0	0
$601$	746.618i	1.24229i	0.783694	+	0.621147i	$0.213333\pi$
$601$	746.618i	1.24229i	−0.783694	+	0.621147i	$0.786667\pi$
$602$	1176.61i	1.95449i
$603$	0	0
$604$	− 1060.00i	− 1.75496i
$605$	0	0
$606$	0	0
$607$	− 76.8780i	− 0.126652i	−0.997993	−	0.0633262i	$-0.979829\pi$
$607$	− 76.8780i	− 0.126652i	0.997993	−	0.0633262i	$-0.0201709\pi$
$608$	382.127	0.628499
$609$	0	0
$610$	820.678	1.34537
$611$	− 4.80587i	− 0.00786558i
$612$	0	0
$613$	462.789i	0.754957i	0.926018	+	0.377479i	$0.123209\pi$
$613$	462.789i	0.754957i	−0.926018	+	0.377479i	$0.876791\pi$
$614$	7.93839	0.0129290
$615$	0	0
$616$	0	0
$617$	−517.342	−0.838479	−0.419240	−	0.907876i	$-0.637703\pi$
$617$	−517.342	−0.838479	−0.419240	+	0.907876i	$0.637703\pi$
$618$	0	0
$619$	−946.440	−1.52898	−0.764491	−	0.644634i	$-0.777010\pi$
$619$	−946.440	−1.52898	−0.764491	+	0.644634i	$0.777010\pi$
$620$	1803.87	2.90947
$621$	0	0
$622$	1160.08i	1.86508i
$623$	− 851.110i	− 1.36615i
$624$	0	0
$625$	−723.814	−1.15810
$626$	− 367.453i	− 0.586985i
$627$	0	0
$628$	−2492.13	−3.96836
$629$	− 901.240i	− 1.43281i
$630$	0	0
$631$	469.930	0.744739	0.372369	−	0.928085i	$-0.378545\pi$
$631$	469.930	0.744739	0.372369	+	0.928085i	$0.378545\pi$
$632$	−1739.23	−2.75194
$633$	0	0
$634$	− 1018.83i	− 1.60698i
$635$	426.475i	0.671615i
$636$	0	0
$637$	15.0016i	0.0235503i
$638$	0	0
$639$	0	0
$640$	− 381.480i	− 0.596062i
$641$	129.811	0.202514	0.101257	−	0.994860i	$-0.467714\pi$
$641$	129.811	0.202514	0.101257	+	0.994860i	$0.467714\pi$
$642$	0	0
$643$	74.9361	0.116541	0.0582707	−	0.998301i	$-0.481441\pi$
$643$	74.9361	0.116541	0.0582707	+	0.998301i	$0.481441\pi$
$644$	1183.35i	1.83750i
$645$	0	0
$646$	568.077i	0.879377i
$647$	−906.156	−1.40055	−0.700275	−	0.713873i	$-0.746939\pi$
$647$	−906.156	−1.40055	−0.700275	+	0.713873i	$0.746939\pi$
$648$	0	0
$649$	0	0
$650$	118.574	0.182421
$651$	0	0
$652$	−2241.73	−3.43824
$653$	1053.26	1.61296	0.806480	−	0.591261i	$-0.201370\pi$
$653$	1053.26	1.61296	0.806480	+	0.591261i	$0.201370\pi$
$654$	0	0
$655$	− 410.035i	− 0.626008i
$656$	1067.25i	1.62690i
$657$	0	0
$658$	19.0399	0.0289360
$659$	− 569.208i	− 0.863746i	−0.901935	−	0.431873i	$-0.857853\pi$
$659$	− 569.208i	− 0.863746i	0.901935	−	0.431873i	$-0.142147\pi$
$660$	0	0
$661$	−436.561	−0.660455	−0.330228	−	0.943901i	$-0.607126\pi$
$661$	−436.561	−0.660455	−0.330228	+	0.943901i	$0.607126\pi$
$662$	− 236.896i	− 0.357850i
$663$	0	0
$664$	510.343	0.768589
$665$	330.146	0.496460
$666$	0	0
$667$	− 355.614i	− 0.533154i
$668$	− 1645.21i	− 2.46290i
$669$	0	0
$670$	− 1918.15i	− 2.86290i
$671$	0	0
$672$	0	0
$673$	− 1225.49i	− 1.82093i	−0.413585	−	0.910465i	$-0.635724\pi$
$673$	− 1225.49i	− 1.82093i	0.413585	−	0.910465i	$-0.364276\pi$
$674$	−2035.16	−3.01953
$675$	0	0
$676$	−1170.57	−1.73161
$677$	− 600.605i	− 0.887156i	−0.896236	−	0.443578i	$-0.853709\pi$
$677$	− 600.605i	− 0.887156i	0.896236	−	0.443578i	$-0.146291\pi$
$678$	0	0
$679$	237.040i	0.349102i
$680$	−1965.24	−2.89006
$681$	0	0
$682$	0	0
$683$	653.422	0.956693	0.478347	−	0.878171i	$-0.341236\pi$
$683$	653.422	0.956693	0.478347	+	0.878171i	$0.341236\pi$
$684$	0	0
$685$	866.596	1.26510
$686$	1221.63	1.78079
$687$	0	0
$688$	1518.45i	2.20705i
$689$	463.113i	0.672153i
$690$	0	0
$691$	−963.085	−1.39375	−0.696877	−	0.717190i	$-0.745428\pi$
$691$	−963.085	−1.39375	−0.696877	+	0.717190i	$0.745428\pi$
$692$	2489.43i	3.59744i
$693$	0	0
$694$	2393.14	3.44833
$695$	− 1412.21i	− 2.03196i
$696$	0	0
$697$	−583.892	−0.837722
$698$	−59.2961	−0.0849514
$699$	0	0
$700$	328.809i	0.469727i
$701$	− 324.852i	− 0.463412i	−0.972786	−	0.231706i	$-0.925569\pi$
$701$	− 324.852i	− 0.463412i	0.972786	−	0.231706i	$-0.0744307\pi$
$702$	0	0
$703$	411.528i	0.585388i
$704$	0	0
$705$	0	0
$706$	1359.11i	1.92508i
$707$	−367.418	−0.519686
$708$	0	0
$709$	1131.05	1.59527	0.797636	−	0.603140i	$-0.206084\pi$
$709$	1131.05	1.59527	0.797636	+	0.603140i	$0.206084\pi$
$710$	790.781i	1.11378i
$711$	0	0
$712$	− 2313.42i	− 3.24919i
$713$	625.967	0.877933
$714$	0	0
$715$	0	0
$716$	−540.556	−0.754967
$717$	0	0
$718$	211.672	0.294807
$719$	−741.112	−1.03075	−0.515377	−	0.856964i	$-0.672348\pi$
$719$	−741.112	−1.03075	−0.515377	+	0.856964i	$0.672348\pi$
$720$	0	0
$721$	− 866.460i	− 1.20175i
$722$	1058.66i	1.46629i
$723$	0	0
$724$	2169.66	2.99677
$725$	− 98.8116i	− 0.136292i
$726$	0	0
$727$	1305.90	1.79629	0.898144	−	0.439702i	$-0.144916\pi$
$727$	1305.90	1.79629	0.898144	+	0.439702i	$0.144916\pi$
$728$	919.689i	1.26331i
$729$	0	0
$730$	1403.21	1.92220
$731$	−830.747	−1.13645
$732$	0	0
$733$	− 253.300i	− 0.345567i	−0.984960	−	0.172783i	$-0.944724\pi$
$733$	− 253.300i	− 0.345567i	0.984960	−	0.172783i	$-0.0552761\pi$
$734$	− 686.715i	− 0.935579i
$735$	0	0
$736$	802.954i	1.09097i
$737$	0	0
$738$	0	0
$739$	− 194.763i	− 0.263549i	−0.991280	−	0.131774i	$-0.957933\pi$
$739$	− 194.763i	− 0.263549i	0.991280	−	0.131774i	$-0.0420674\pi$
$740$	−2491.93	−3.36748
$741$	0	0
$742$	−1834.76	−2.47273
$743$	− 158.712i	− 0.213610i	−0.994280	−	0.106805i	$-0.965938\pi$
$743$	− 158.712i	− 0.213610i	0.994280	−	0.106805i	$-0.0340621\pi$
$744$	0	0
$745$	− 307.545i	− 0.412813i
$746$	−1719.33	−2.30473
$747$	0	0
$748$	0	0
$749$	1270.83	1.69670
$750$	0	0
$751$	−127.834	−0.170218	−0.0851089	−	0.996372i	$-0.527124\pi$
$751$	−127.834	−0.170218	−0.0851089	+	0.996372i	$0.527124\pi$
$752$	24.5717	0.0326751
$753$	0	0
$754$	− 483.764i	− 0.641597i
$755$	− 621.410i	− 0.823059i
$756$	0	0
$757$	−1209.45	−1.59769	−0.798843	−	0.601539i	$-0.794554\pi$
$757$	−1209.45	−1.59769	−0.798843	+	0.601539i	$0.794554\pi$
$758$	1033.13i	1.36297i
$759$	0	0
$760$	897.377	1.18076
$761$	− 220.256i	− 0.289430i	−0.989473	−	0.144715i	$-0.953773\pi$
$761$	− 220.256i	− 0.289430i	0.989473	−	0.144715i	$-0.0462266\pi$
$762$	0	0
$763$	584.473	0.766020
$764$	127.841	0.167332
$765$	0	0
$766$	− 252.668i	− 0.329853i
$767$	159.078i	0.207403i
$768$	0	0
$769$	− 843.244i	− 1.09655i	−0.836299	−	0.548273i	$-0.815285\pi$
$769$	− 843.244i	− 1.09655i	0.836299	−	0.548273i	$-0.184715\pi$
$770$	0	0
$771$	0	0
$772$	− 322.296i	− 0.417482i
$773$	112.545	0.145595	0.0727974	−	0.997347i	$-0.476807\pi$
$773$	112.545	0.145595	0.0727974	+	0.997347i	$0.476807\pi$
$774$	0	0
$775$	173.932	0.224429
$776$	644.305i	0.830289i
$777$	0	0
$778$	2156.76i	2.77218i
$779$	266.619	0.342258
$780$	0	0
$781$	0	0
$782$	−1193.69	−1.52645
$783$	0	0
$784$	−76.7007	−0.0978325
$785$	−1460.98	−1.86112
$786$	0	0
$787$	975.006i	1.23889i	0.785040	+	0.619445i	$0.212642\pi$
$787$	975.006i	1.23889i	−0.785040	+	0.619445i	$0.787358\pi$
$788$	− 969.538i	− 1.23038i
$789$	0	0
$790$	−1784.68	−2.25908
$791$	− 153.179i	− 0.193653i
$792$	0	0
$793$	271.166	0.341950
$794$	− 1796.79i	− 2.26296i
$795$	0	0
$796$	−2470.93	−3.10419
$797$	1479.65	1.85653	0.928263	−	0.371926i	$-0.121302\pi$
$797$	1479.65	1.85653	0.928263	+	0.371926i	$0.121302\pi$
$798$	0	0
$799$	13.4432i	0.0168250i
$800$	223.111i	0.278888i
$801$	0	0
$802$	1986.48i	2.47691i
$803$	0	0
$804$	0	0
$805$	693.727i	0.861772i
$806$	851.542	1.05650
$807$	0	0
$808$	−998.688	−1.23600
$809$	164.996i	0.203951i	0.994787	+	0.101975i	$0.0325163\pi$
$809$	164.996i	0.203951i	−0.994787	+	0.101975i	$0.967484\pi$
$810$	0	0
$811$	− 837.213i	− 1.03232i	−0.856492	−	0.516161i	$-0.827361\pi$
$811$	− 837.213i	− 1.03232i	0.856492	−	0.516161i	$-0.172639\pi$
$812$	1341.49	1.65208
$813$	0	0
$814$	0	0
$815$	−1314.19	−1.61250
$816$	0	0
$817$	379.339	0.464307
$818$	976.145	1.19333
$819$	0	0
$820$	1614.46i	1.96886i
$821$	− 1285.24i	− 1.56546i	−0.622361	−	0.782730i	$-0.713827\pi$
$821$	− 1285.24i	− 1.56546i	0.622361	−	0.782730i	$-0.286173\pi$
$822$	0	0
$823$	729.863	0.886833	0.443416	−	0.896316i	$-0.353766\pi$
$823$	729.863	0.886833	0.443416	+	0.896316i	$0.353766\pi$
$824$	− 2355.15i	− 2.85819i
$825$	0	0
$826$	−630.237	−0.762998
$827$	535.281i	0.647257i	0.946184	+	0.323628i	$0.104903\pi$
$827$	535.281i	0.647257i	−0.946184	+	0.323628i	$0.895097\pi$
$828$	0	0
$829$	620.932	0.749013	0.374506	−	0.927224i	$-0.377812\pi$
$829$	620.932	0.749013	0.374506	+	0.927224i	$0.377812\pi$
$830$	523.678	0.630938
$831$	0	0
$832$	201.709i	0.242439i
$833$	− 41.9631i	− 0.0503758i
$834$	0	0
$835$	− 964.487i	− 1.15507i
$836$	0	0
$837$	0	0
$838$	− 15.7528i	− 0.0187981i
$839$	−845.197	−1.00739	−0.503693	−	0.863883i	$-0.668026\pi$
$839$	−845.197	−1.00739	−0.503693	+	0.863883i	$0.668026\pi$
$840$	0	0
$841$	437.863	0.520646
$842$	− 1184.92i	− 1.40726i
$843$	0	0
$844$	− 2752.45i	− 3.26120i
$845$	−686.232	−0.812108
$846$	0	0
$847$	0	0
$848$	−2367.83	−2.79225
$849$	0	0
$850$	−331.680	−0.390212
$851$	−864.733	−1.01614
$852$	0	0
$853$	− 369.694i	− 0.433404i	−0.976238	−	0.216702i	$-0.930470\pi$
$853$	− 369.694i	− 0.433404i	0.976238	−	0.216702i	$-0.0695301\pi$
$854$	1074.31i	1.25797i
$855$	0	0
$856$	3454.27	4.03536
$857$	− 58.2378i	− 0.0679554i	−0.999423	−	0.0339777i	$-0.989182\pi$
$857$	− 58.2378i	− 0.0679554i	0.999423	−	0.0339777i	$-0.0108175\pi$
$858$	0	0
$859$	−708.720	−0.825052	−0.412526	−	0.910946i	$-0.635353\pi$
$859$	−708.720	−0.825052	−0.412526	+	0.910946i	$0.635353\pi$
$860$	2297.02i	2.67095i
$861$	0	0
$862$	−287.315	−0.333312
$863$	121.657	0.140970	0.0704851	−	0.997513i	$-0.477545\pi$
$863$	121.657	0.140970	0.0704851	+	0.997513i	$0.477545\pi$
$864$	0	0
$865$	1459.40i	1.68716i
$866$	− 1262.77i	− 1.45817i
$867$	0	0
$868$	2361.35i	2.72045i
$869$	0	0
$870$	0	0
$871$	− 633.788i	− 0.727656i
$872$	1588.67	1.82187
$873$	0	0
$874$	545.066	0.623646
$875$	− 786.450i	− 0.898800i
$876$	0	0
$877$	48.4114i	0.0552011i	0.999619	+	0.0276006i	$0.00878665\pi$
$877$	48.4114i	0.0552011i	−0.999619	+	0.0276006i	$0.991213\pi$
$878$	1390.28	1.58346
$879$	0	0
$880$	0	0
$881$	−820.445	−0.931265	−0.465633	−	0.884978i	$-0.654173\pi$
$881$	−820.445	−0.931265	−0.465633	+	0.884978i	$0.654173\pi$
$882$	0	0
$883$	−10.7683	−0.0121951	−0.00609755	−	0.999981i	$-0.501941\pi$
$883$	−10.7683	−0.0121951	−0.00609755	+	0.999981i	$0.501941\pi$
$884$	−1136.60	−1.28575
$885$	0	0
$886$	603.213i	0.680828i
$887$	− 648.187i	− 0.730763i	−0.930858	−	0.365381i	$-0.880939\pi$
$887$	− 648.187i	− 0.730763i	0.930858	−	0.365381i	$-0.119061\pi$
$888$	0	0
$889$	−558.276	−0.627982
$890$	− 2373.88i	− 2.66728i
$891$	0	0
$892$	−1929.78	−2.16343
$893$	− 6.13849i	− 0.00687401i
$894$	0	0
$895$	−316.895	−0.354072
$896$	499.375	0.557338
$897$	0	0
$898$	1698.19i	1.89108i
$899$	− 709.619i	− 0.789342i
$900$	0	0
$901$	− 1295.44i	− 1.43778i
$902$	0	0
$903$	0	0
$904$	− 416.361i	− 0.460576i
$905$	1271.94	1.40546
$906$	0	0
$907$	590.053	0.650555	0.325277	−	0.945619i	$-0.394542\pi$
$907$	590.053	0.650555	0.325277	+	0.945619i	$0.394542\pi$
$908$	2695.59i	2.96871i
$909$	0	0
$910$	943.720i	1.03706i
$911$	1251.15	1.37338	0.686692	−	0.726949i	$-0.259062\pi$
$911$	1251.15	1.37338	0.686692	+	0.726949i	$0.259062\pi$
$912$	0	0
$913$	0	0
$914$	2585.10	2.82834
$915$	0	0
$916$	3596.14	3.92592
$917$	536.756	0.585339
$918$	0	0
$919$	278.871i	0.303450i	0.988423	+	0.151725i	$0.0484829\pi$
$919$	278.871i	0.303450i	−0.988423	+	0.151725i	$0.951517\pi$
$920$	1885.64i	2.04960i
$921$	0	0
$922$	547.515	0.593834
$923$	261.288i	0.283085i
$924$	0	0
$925$	−240.277	−0.259758
$926$	624.324i	0.674216i
$927$	0	0
$928$	910.258	0.980882
$929$	433.213	0.466322	0.233161	−	0.972438i	$-0.425093\pi$
$929$	433.213	0.466322	0.233161	+	0.972438i	$0.425093\pi$
$930$	0	0
$931$	19.1614i	0.0205815i
$932$	1323.79i	1.42037i
$933$	0	0
$934$	2409.24i	2.57948i
$935$	0	0
$936$	0	0
$937$	1105.41i	1.17974i	0.807499	+	0.589868i	$0.200820\pi$
$937$	1105.41i	1.17974i	−0.807499	+	0.589868i	$0.799180\pi$
$938$	2510.94	2.67691
$939$	0	0
$940$	37.1705	0.0395431
$941$	− 49.8081i	− 0.0529311i	−0.999650	−	0.0264655i	$-0.991575\pi$
$941$	− 49.8081i	− 0.0529311i	0.999650	−	0.0264655i	$-0.00842522\pi$
$942$	0	0
$943$	560.240i	0.594104i
$944$	−813.343	−0.861592
$945$	0	0
$946$	0	0
$947$	−643.764	−0.679793	−0.339896	−	0.940463i	$-0.610392\pi$
$947$	−643.764	−0.679793	−0.339896	+	0.940463i	$0.610392\pi$
$948$	0	0
$949$	463.643	0.488560
$950$	151.453	0.159424
$951$	0	0
$952$	− 2572.60i	− 2.70231i
$953$	1356.65i	1.42356i	0.702403	+	0.711779i	$0.252110\pi$
$953$	1356.65i	1.42356i	−0.702403	+	0.711779i	$0.747890\pi$
$954$	0	0
$955$	74.9455	0.0784770
$956$	1872.31i	1.95848i
$957$	0	0
$958$	−945.203	−0.986642
$959$	1134.42i	1.18291i
$960$	0	0
$961$	288.101	0.299792
$962$	−1176.35	−1.22282
$963$	0	0
$964$	− 1432.77i	− 1.48627i
$965$	− 188.942i	− 0.195795i
$966$	0	0
$967$	863.704i	0.893179i	0.894739	+	0.446589i	$0.147362\pi$
$967$	863.704i	0.893179i	−0.894739	+	0.446589i	$0.852638\pi$
$968$	0	0
$969$	0	0
$970$	661.140i	0.681588i
$971$	−1040.49	−1.07156	−0.535781	−	0.844357i	$-0.679983\pi$
$971$	−1040.49	−1.07156	−0.535781	+	0.844357i	$0.679983\pi$
$972$	0	0
$973$	1848.66	1.89995
$974$	− 193.565i	− 0.198732i
$975$	0	0
$976$	1386.43i	1.42052i
$977$	893.176	0.914203	0.457101	−	0.889415i	$-0.348888\pi$
$977$	893.176	0.914203	0.457101	+	0.889415i	$0.348888\pi$
$978$	0	0
$979$	0	0
$980$	−116.028	−0.118396
$981$	0	0
$982$	−1246.39	−1.26924
$983$	−190.326	−0.193617	−0.0968087	−	0.995303i	$-0.530864\pi$
$983$	−190.326	−0.193617	−0.0968087	+	0.995303i	$0.530864\pi$
$984$	0	0
$985$	− 568.380i	− 0.577036i
$986$	1353.21i	1.37242i
$987$	0	0
$988$	518.998	0.525302
$989$	797.095i	0.805961i
$990$	0	0
$991$	−139.682	−0.140950	−0.0704751	−	0.997514i	$-0.522452\pi$
$991$	−139.682	−0.140950	−0.0704751	+	0.997514i	$0.522452\pi$
$992$	1602.27i	1.61520i
$993$	0	0
$994$	−1035.17	−1.04142
$995$	−1448.55	−1.45583
$996$	0	0
$997$	− 209.787i	− 0.210418i	−0.994450	−	0.105209i	$-0.966449\pi$
$997$	− 209.787i	− 0.210418i	0.994450	−	0.105209i	$-0.0335511\pi$
$998$	− 640.292i	− 0.641575i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.m.604.16		16
3.2	odd	2		363.3.c.e.241.1		16
11.3	even	5		99.3.k.c.46.4		16
11.7	odd	10		99.3.k.c.28.4		16
11.10	odd	2	inner	1089.3.c.m.604.1		16
33.2	even	10		363.3.g.g.40.4		16
33.5	odd	10		363.3.g.g.118.4		16
33.8	even	10		363.3.g.f.112.4		16
33.14	odd	10		33.3.g.a.13.1	✓	16
33.17	even	10		363.3.g.a.118.1		16
33.20	odd	10		363.3.g.a.40.1		16
33.26	odd	10		363.3.g.f.94.4		16
33.29	even	10		33.3.g.a.28.1	yes	16
33.32	even	2		363.3.c.e.241.16		16
132.47	even	10		528.3.bf.b.145.4		16
132.95	odd	10		528.3.bf.b.193.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.13.1	✓	16	33.14	odd	10
33.3.g.a.28.1	yes	16	33.29	even	10
99.3.k.c.28.4		16	11.7	odd	10
99.3.k.c.46.4		16	11.3	even	5
363.3.c.e.241.1		16	3.2	odd	2
363.3.c.e.241.16		16	33.32	even	2
363.3.g.a.40.1		16	33.20	odd	10
363.3.g.a.118.1		16	33.17	even	10
363.3.g.f.94.4		16	33.26	odd	10
363.3.g.f.112.4		16	33.8	even	10
363.3.g.g.40.4		16	33.2	even	10
363.3.g.g.118.4		16	33.5	odd	10
528.3.bf.b.145.4		16	132.47	even	10
528.3.bf.b.193.4		16	132.95	odd	10
1089.3.c.m.604.1		16	11.10	odd	2	inner
1089.3.c.m.604.16		16	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.65113i q^{2} -9.33074 q^{4} -5.47004 q^{5} -7.16054i q^{7} -19.4632i q^{8} +O(q^{10})\)	\(q+3.65113i q^{2} -9.33074 q^{4} -5.47004 q^{5} -7.16054i q^{7} -19.4632i q^{8} -19.9718i q^{10} -6.59903i q^{13} +26.1440 q^{14} +33.7398 q^{16} +18.4591i q^{17} -8.42888i q^{19} +51.0395 q^{20} +17.7114 q^{23} +4.92132 q^{25} +24.0939 q^{26} +66.8132i q^{28} -20.0783i q^{29} +35.3426 q^{31} +45.3355i q^{32} -67.3966 q^{34} +39.1684i q^{35} -48.8236 q^{37} +30.7749 q^{38} +106.465i q^{40} +31.6316i q^{41} +45.0047i q^{43} +64.6665i q^{46} +0.728269 q^{47} -2.27330 q^{49} +17.9684i q^{50} +61.5739i q^{52} -70.1790 q^{53} -139.367 q^{56} +73.3084 q^{58} -24.1063 q^{59} +41.0918i q^{61} +129.040i q^{62} -30.5665 q^{64} +36.0970i q^{65} +96.0426 q^{67} -172.237i q^{68} -143.009 q^{70} -39.5948 q^{71} +70.2593i q^{73} -178.261i q^{74} +78.6477i q^{76} -89.3597i q^{79} -184.558 q^{80} -115.491 q^{82} +26.2209i q^{83} -100.972i q^{85} -164.318 q^{86} +118.861 q^{89} -47.2526 q^{91} -165.260 q^{92} +2.65900i q^{94} +46.1063i q^{95} -33.1037 q^{97} -8.30011i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) \) 16 * q - 20 * q^4 + 4 * q^5	\( 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) \) 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more