LMFDB - Embedded newform 1089.3.b.e.485.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(485,1089)]

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([1, 0]))

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

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

chi := DirichletCharacter("1089.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.b (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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.2
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.e.485.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949i q^{2} -2.00000 q^{4} +8.48528i q^{5} +12.1244 q^{7} -4.89898i q^{8} +O(q^{10})$	$q-2.44949i q^{2} -2.00000 q^{4} +8.48528i q^{5} +12.1244 q^{7} -4.89898i q^{8} +20.7846 q^{10} +3.46410 q^{13} -29.6985i q^{14} -20.0000 q^{16} -24.4949i q^{17} +1.73205 q^{19} -16.9706i q^{20} -4.24264i q^{23} -47.0000 q^{25} -8.48528i q^{26} -24.2487 q^{28} +31.8434i q^{29} +37.0000 q^{31} +29.3939i q^{32} -60.0000 q^{34} +102.879i q^{35} +61.0000 q^{37} -4.24264i q^{38} +41.5692 q^{40} -41.6413i q^{41} +45.0333 q^{43} -10.3923 q^{46} +29.6985i q^{47} +98.0000 q^{49} +115.126i q^{50} -6.92820 q^{52} +4.24264i q^{53} -59.3970i q^{56} +78.0000 q^{58} -21.2132i q^{59} -15.5885 q^{61} -90.6311i q^{62} -8.00000 q^{64} +29.3939i q^{65} -41.0000 q^{67} +48.9898i q^{68} +252.000 q^{70} +76.3675i q^{71} -60.6218 q^{73} -149.419i q^{74} -3.46410 q^{76} +129.904 q^{79} -169.706i q^{80} -102.000 q^{82} +95.5301i q^{83} +207.846 q^{85} -110.309i q^{86} -50.9117i q^{89} +42.0000 q^{91} +8.48528i q^{92} +72.7461 q^{94} +14.6969i q^{95} +179.000 q^{97} -240.050i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} - 80 q^{16} - 188 q^{25} + 148 q^{31} - 240 q^{34} + 244 q^{37} + 392 q^{49} + 312 q^{58} - 32 q^{64} - 164 q^{67} + 1008 q^{70} - 408 q^{82} + 168 q^{91} + 716 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 - 80 * q^16 - 188 * q^25 + 148 * q^31 - 240 * q^34 + 244 * q^37 + 392 * q^49 + 312 * q^58 - 32 * q^64 - 164 * q^67 + 1008 * q^70 - 408 * q^82 + 168 * q^91 + 716 * 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 $

$2$	− 2.44949i	− 1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	− 2.44949i	− 1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	8.48528i	1.69706i	0.529150	+	0.848528i	$0.322511\pi$
$5$	8.48528i	1.69706i	−0.529150	+	0.848528i	$0.677489\pi$
$6$	0	0
$7$	12.1244	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$7$	12.1244	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$8$	− 4.89898i	− 0.612372i
$9$	0	0
$10$	20.7846	2.07846
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.46410	0.266469	0.133235	−	0.991085i	$-0.457464\pi$
$13$	3.46410	0.266469	0.133235	+	0.991085i	$0.457464\pi$
$14$	− 29.6985i	− 2.12132i
$15$	0	0
$16$	−20.0000	−1.25000
$17$	− 24.4949i	− 1.44088i	−0.693519	−	0.720438i	$-0.743941\pi$
$17$	− 24.4949i	− 1.44088i	0.693519	−	0.720438i	$-0.256059\pi$
$18$	0	0
$19$	1.73205	0.0911606	0.0455803	−	0.998961i	$-0.485486\pi$
$19$	1.73205	0.0911606	0.0455803	+	0.998961i	$0.485486\pi$
$20$	− 16.9706i	− 0.848528i
$21$	0	0
$22$	0	0
$23$	− 4.24264i	− 0.184463i	−0.995738	−	0.0922313i	$-0.970600\pi$
$23$	− 4.24264i	− 0.184463i	0.995738	−	0.0922313i	$-0.0293999\pi$
$24$	0	0
$25$	−47.0000	−1.88000
$26$	− 8.48528i	− 0.326357i
$27$	0	0
$28$	−24.2487	−0.866025
$29$	31.8434i	1.09805i	0.835807	+	0.549024i	$0.185000\pi$
$29$	31.8434i	1.09805i	−0.835807	+	0.549024i	$0.815000\pi$
$30$	0	0
$31$	37.0000	1.19355	0.596774	−	0.802409i	$-0.296449\pi$
$31$	37.0000	1.19355	0.596774	+	0.802409i	$0.296449\pi$
$32$	29.3939i	0.918559i
$33$	0	0
$34$	−60.0000	−1.76471
$35$	102.879i	2.93939i
$36$	0	0
$37$	61.0000	1.64865	0.824324	−	0.566118i	$-0.191555\pi$
$37$	61.0000	1.64865	0.824324	+	0.566118i	$0.191555\pi$
$38$	− 4.24264i	− 0.111648i
$39$	0	0
$40$	41.5692	1.03923
$41$	− 41.6413i	− 1.01564i	−0.861463	−	0.507821i	$-0.830451\pi$
$41$	− 41.6413i	− 1.01564i	0.861463	−	0.507821i	$-0.169549\pi$
$42$	0	0
$43$	45.0333	1.04729	0.523643	−	0.851938i	$-0.324572\pi$
$43$	45.0333	1.04729	0.523643	+	0.851938i	$0.324572\pi$
$44$	0	0
$45$	0	0
$46$	−10.3923	−0.225920
$47$	29.6985i	0.631883i	0.948779	+	0.315941i	$0.102320\pi$
$47$	29.6985i	0.631883i	−0.948779	+	0.315941i	$0.897680\pi$
$48$	0	0
$49$	98.0000	2.00000
$50$	115.126i	2.30252i
$51$	0	0
$52$	−6.92820	−0.133235
$53$	4.24264i	0.0800498i	0.999199	+	0.0400249i	$0.0127437\pi$
$53$	4.24264i	0.0800498i	−0.999199	+	0.0400249i	$0.987256\pi$
$54$	0	0
$55$	0	0
$56$	− 59.3970i	− 1.06066i
$57$	0	0
$58$	78.0000	1.34483
$59$	− 21.2132i	− 0.359546i	−0.983708	−	0.179773i	$-0.942464\pi$
$59$	− 21.2132i	− 0.359546i	0.983708	−	0.179773i	$-0.0575363\pi$
$60$	0	0
$61$	−15.5885	−0.255548	−0.127774	−	0.991803i	$-0.540783\pi$
$61$	−15.5885	−0.255548	−0.127774	+	0.991803i	$0.540783\pi$
$62$	− 90.6311i	− 1.46179i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	29.3939i	0.452213i
$66$	0	0
$67$	−41.0000	−0.611940	−0.305970	−	0.952041i	$-0.598981\pi$
$67$	−41.0000	−0.611940	−0.305970	+	0.952041i	$0.598981\pi$
$68$	48.9898i	0.720438i
$69$	0	0
$70$	252.000	3.60000
$71$	76.3675i	1.07560i	0.843073	+	0.537800i	$0.180744\pi$
$71$	76.3675i	1.07560i	−0.843073	+	0.537800i	$0.819256\pi$
$72$	0	0
$73$	−60.6218	−0.830435	−0.415218	−	0.909722i	$-0.636295\pi$
$73$	−60.6218	−0.830435	−0.415218	+	0.909722i	$0.636295\pi$
$74$	− 149.419i	− 2.01917i
$75$	0	0
$76$	−3.46410	−0.0455803
$77$	0	0
$78$	0	0
$79$	129.904	1.64435	0.822176	−	0.569233i	$-0.192760\pi$
$79$	129.904	1.64435	0.822176	+	0.569233i	$0.192760\pi$
$80$	− 169.706i	− 2.12132i
$81$	0	0
$82$	−102.000	−1.24390
$83$	95.5301i	1.15097i	0.817814	+	0.575483i	$0.195186\pi$
$83$	95.5301i	1.15097i	−0.817814	+	0.575483i	$0.804814\pi$
$84$	0	0
$85$	207.846	2.44525
$86$	− 110.309i	− 1.28266i
$87$	0	0
$88$	0	0
$89$	− 50.9117i	− 0.572041i	−0.958223	−	0.286021i	$-0.907667\pi$
$89$	− 50.9117i	− 0.572041i	0.958223	−	0.286021i	$-0.0923326\pi$
$90$	0	0
$91$	42.0000	0.461538
$92$	8.48528i	0.0922313i
$93$	0	0
$94$	72.7461	0.773895
$95$	14.6969i	0.154705i
$96$	0	0
$97$	179.000	1.84536	0.922680	−	0.385566i	$-0.125994\pi$
$97$	179.000	1.84536	0.922680	+	0.385566i	$0.125994\pi$
$98$	− 240.050i	− 2.44949i
$99$	0	0
$100$	94.0000	0.940000
$101$	41.6413i	0.412290i	0.978521	+	0.206145i	$0.0660919\pi$
$101$	41.6413i	0.412290i	−0.978521	+	0.206145i	$0.933908\pi$
$102$	0	0
$103$	−145.000	−1.40777	−0.703883	−	0.710315i	$-0.748552\pi$
$103$	−145.000	−1.40777	−0.703883	+	0.710315i	$0.748552\pi$
$104$	− 16.9706i	− 0.163178i
$105$	0	0
$106$	10.3923	0.0980406
$107$	46.5403i	0.434956i	0.976065	+	0.217478i	$0.0697831\pi$
$107$	46.5403i	0.434956i	−0.976065	+	0.217478i	$0.930217\pi$
$108$	0	0
$109$	46.7654	0.429040	0.214520	−	0.976720i	$-0.431181\pi$
$109$	46.7654	0.429040	0.214520	+	0.976720i	$0.431181\pi$
$110$	0	0
$111$	0	0
$112$	−242.487	−2.16506
$113$	38.1838i	0.337909i	0.985624	+	0.168955i	$0.0540392\pi$
$113$	38.1838i	0.337909i	−0.985624	+	0.168955i	$0.945961\pi$
$114$	0	0
$115$	36.0000	0.313043
$116$	− 63.6867i	− 0.549024i
$117$	0	0
$118$	−51.9615	−0.440352
$119$	− 296.985i	− 2.49567i
$120$	0	0
$121$	0	0
$122$	38.1838i	0.312982i
$123$	0	0
$124$	−74.0000	−0.596774
$125$	− 186.676i	− 1.49341i
$126$	0	0
$127$	−171.473	−1.35018	−0.675091	−	0.737735i	$-0.735896\pi$
$127$	−171.473	−1.35018	−0.675091	+	0.737735i	$0.735896\pi$
$128$	137.171i	1.07165i
$129$	0	0
$130$	72.0000	0.553846
$131$	61.2372i	0.467460i	0.972302	+	0.233730i	$0.0750932\pi$
$131$	61.2372i	0.467460i	−0.972302	+	0.233730i	$0.924907\pi$
$132$	0	0
$133$	21.0000	0.157895
$134$	100.429i	0.749471i
$135$	0	0
$136$	−120.000	−0.882353
$137$	127.279i	0.929045i	0.885561	+	0.464523i	$0.153774\pi$
$137$	127.279i	0.929045i	−0.885561	+	0.464523i	$0.846226\pi$
$138$	0	0
$139$	83.1384	0.598118	0.299059	−	0.954235i	$-0.403327\pi$
$139$	83.1384	0.598118	0.299059	+	0.954235i	$0.403327\pi$
$140$	− 205.757i	− 1.46969i
$141$	0	0
$142$	187.061	1.31733
$143$	0	0
$144$	0	0
$145$	−270.200	−1.86345
$146$	148.492i	1.01707i
$147$	0	0
$148$	−122.000	−0.824324
$149$	− 19.5959i	− 0.131516i	−0.997836	−	0.0657581i	$-0.979053\pi$
$149$	− 19.5959i	− 0.131516i	0.997836	−	0.0657581i	$-0.0209466\pi$
$150$	0	0
$151$	3.46410	0.0229411	0.0114705	−	0.999934i	$-0.496349\pi$
$151$	3.46410	0.0229411	0.0114705	+	0.999934i	$0.496349\pi$
$152$	− 8.48528i	− 0.0558242i
$153$	0	0
$154$	0	0
$155$	313.955i	2.02552i
$156$	0	0
$157$	127.000	0.808917	0.404459	−	0.914556i	$-0.367460\pi$
$157$	127.000	0.808917	0.404459	+	0.914556i	$0.367460\pi$
$158$	− 318.198i	− 2.01391i
$159$	0	0
$160$	−249.415	−1.55885
$161$	− 51.4393i	− 0.319499i
$162$	0	0
$163$	−133.000	−0.815951	−0.407975	−	0.912993i	$-0.633765\pi$
$163$	−133.000	−0.815951	−0.407975	+	0.912993i	$0.633765\pi$
$164$	83.2827i	0.507821i
$165$	0	0
$166$	234.000	1.40964
$167$	− 252.297i	− 1.51076i	−0.655285	−	0.755382i	$-0.727452\pi$
$167$	− 252.297i	− 1.51076i	0.655285	−	0.755382i	$-0.272548\pi$
$168$	0	0
$169$	−157.000	−0.928994
$170$	− 509.117i	− 2.99481i
$171$	0	0
$172$	−90.0666	−0.523643
$173$	− 44.0908i	− 0.254860i	−0.991848	−	0.127430i	$-0.959327\pi$
$173$	− 44.0908i	− 0.254860i	0.991848	−	0.127430i	$-0.0406728\pi$
$174$	0	0
$175$	−569.845	−3.25626
$176$	0	0
$177$	0	0
$178$	−124.708	−0.700605
$179$	− 275.772i	− 1.54062i	−0.637667	−	0.770312i	$-0.720101\pi$
$179$	− 275.772i	− 1.54062i	0.637667	−	0.770312i	$-0.279899\pi$
$180$	0	0
$181$	−187.000	−1.03315	−0.516575	−	0.856242i	$-0.672793\pi$
$181$	−187.000	−1.03315	−0.516575	+	0.856242i	$0.672793\pi$
$182$	− 102.879i	− 0.565267i
$183$	0	0
$184$	−20.7846	−0.112960
$185$	517.602i	2.79785i
$186$	0	0
$187$	0	0
$188$	− 59.3970i	− 0.315941i
$189$	0	0
$190$	36.0000	0.189474
$191$	− 288.500i	− 1.51047i	−0.655455	−	0.755234i	$-0.727523\pi$
$191$	− 288.500i	− 1.51047i	0.655455	−	0.755234i	$-0.272477\pi$
$192$	0	0
$193$	77.9423	0.403846	0.201923	−	0.979401i	$-0.435281\pi$
$193$	77.9423	0.403846	0.201923	+	0.979401i	$0.435281\pi$
$194$	− 438.459i	− 2.26010i
$195$	0	0
$196$	−196.000	−1.00000
$197$	− 171.464i	− 0.870377i	−0.900339	−	0.435189i	$-0.856682\pi$
$197$	− 171.464i	− 0.870377i	0.900339	−	0.435189i	$-0.143318\pi$
$198$	0	0
$199$	−107.000	−0.537688	−0.268844	−	0.963184i	$-0.586642\pi$
$199$	−107.000	−0.537688	−0.268844	+	0.963184i	$0.586642\pi$
$200$	230.252i	1.15126i
$201$	0	0
$202$	102.000	0.504950
$203$	386.080i	1.90187i
$204$	0	0
$205$	353.338	1.72360
$206$	355.176i	1.72416i
$207$	0	0
$208$	−69.2820	−0.333087
$209$	0	0
$210$	0	0
$211$	91.7987	0.435065	0.217532	−	0.976053i	$-0.430199\pi$
$211$	91.7987	0.435065	0.217532	+	0.976053i	$0.430199\pi$
$212$	− 8.48528i	− 0.0400249i
$213$	0	0
$214$	114.000	0.532710
$215$	382.120i	1.77730i
$216$	0	0
$217$	448.601	2.06729
$218$	− 114.551i	− 0.525465i
$219$	0	0
$220$	0	0
$221$	− 84.8528i	− 0.383949i
$222$	0	0
$223$	71.0000	0.318386	0.159193	−	0.987248i	$-0.449111\pi$
$223$	71.0000	0.318386	0.159193	+	0.987248i	$0.449111\pi$
$224$	356.382i	1.59099i
$225$	0	0
$226$	93.5307	0.413853
$227$	151.868i	0.669024i	0.942392	+	0.334512i	$0.108571\pi$
$227$	151.868i	0.669024i	−0.942392	+	0.334512i	$0.891429\pi$
$228$	0	0
$229$	−80.0000	−0.349345	−0.174672	−	0.984627i	$-0.555887\pi$
$229$	−80.0000	−0.349345	−0.174672	+	0.984627i	$0.555887\pi$
$230$	− 88.1816i	− 0.383398i
$231$	0	0
$232$	156.000	0.672414
$233$	− 364.974i	− 1.56641i	−0.621762	−	0.783206i	$-0.713583\pi$
$233$	− 364.974i	− 1.56641i	0.621762	−	0.783206i	$-0.286417\pi$
$234$	0	0
$235$	−252.000	−1.07234
$236$	42.4264i	0.179773i
$237$	0	0
$238$	−727.461	−3.05656
$239$	− 129.823i	− 0.543192i	−0.962411	−	0.271596i	$-0.912449\pi$
$239$	− 129.823i	− 0.543192i	0.962411	−	0.271596i	$-0.0875515\pi$
$240$	0	0
$241$	−51.9615	−0.215608	−0.107804	−	0.994172i	$-0.534382\pi$
$241$	−51.9615	−0.215608	−0.107804	+	0.994172i	$0.534382\pi$
$242$	0	0
$243$	0	0
$244$	31.1769	0.127774
$245$	831.558i	3.39411i
$246$	0	0
$247$	6.00000	0.0242915
$248$	− 181.262i	− 0.730896i
$249$	0	0
$250$	−457.261	−1.82905
$251$	− 182.434i	− 0.726827i	−0.931628	−	0.363413i	$-0.881611\pi$
$251$	− 182.434i	− 0.726827i	0.931628	−	0.363413i	$-0.118389\pi$
$252$	0	0
$253$	0	0
$254$	420.021i	1.65363i
$255$	0	0
$256$	304.000	1.18750
$257$	420.021i	1.63432i	0.576408	+	0.817162i	$0.304454\pi$
$257$	420.021i	1.63432i	−0.576408	+	0.817162i	$0.695546\pi$
$258$	0	0
$259$	739.586	2.85554
$260$	− 58.7878i	− 0.226107i
$261$	0	0
$262$	150.000	0.572519
$263$	134.722i	0.512251i	0.966644	+	0.256125i	$0.0824460\pi$
$263$	134.722i	0.512251i	−0.966644	+	0.256125i	$0.917554\pi$
$264$	0	0
$265$	−36.0000	−0.135849
$266$	− 51.4393i	− 0.193381i
$267$	0	0
$268$	82.0000	0.305970
$269$	− 12.7279i	− 0.0473157i	−0.999720	−	0.0236578i	$-0.992469\pi$
$269$	− 12.7279i	− 0.0473157i	0.999720	−	0.0236578i	$-0.00753123\pi$
$270$	0	0
$271$	−65.8179	−0.242871	−0.121435	−	0.992599i	$-0.538750\pi$
$271$	−65.8179	−0.242871	−0.121435	+	0.992599i	$0.538750\pi$
$272$	489.898i	1.80110i
$273$	0	0
$274$	311.769	1.13784
$275$	0	0
$276$	0	0
$277$	36.3731	0.131311	0.0656554	−	0.997842i	$-0.479086\pi$
$277$	36.3731	0.131311	0.0656554	+	0.997842i	$0.479086\pi$
$278$	− 203.647i	− 0.732542i
$279$	0	0
$280$	504.000	1.80000
$281$	− 313.535i	− 1.11578i	−0.829914	−	0.557891i	$-0.811611\pi$
$281$	− 313.535i	− 1.11578i	0.829914	−	0.557891i	$-0.188389\pi$
$282$	0	0
$283$	−348.142	−1.23018	−0.615092	−	0.788455i	$-0.710881\pi$
$283$	−348.142	−1.23018	−0.615092	+	0.788455i	$0.710881\pi$
$284$	− 152.735i	− 0.537800i
$285$	0	0
$286$	0	0
$287$	− 504.874i	− 1.75914i
$288$	0	0
$289$	−311.000	−1.07612
$290$	661.852i	2.28225i
$291$	0	0
$292$	121.244	0.415218
$293$	421.312i	1.43793i	0.695048	+	0.718963i	$0.255383\pi$
$293$	421.312i	1.43793i	−0.695048	+	0.718963i	$0.744617\pi$
$294$	0	0
$295$	180.000	0.610169
$296$	− 298.838i	− 1.00959i
$297$	0	0
$298$	−48.0000	−0.161074
$299$	− 14.6969i	− 0.0491536i
$300$	0	0
$301$	546.000	1.81395
$302$	− 8.48528i	− 0.0280970i
$303$	0	0
$304$	−34.6410	−0.113951
$305$	− 132.272i	− 0.433680i
$306$	0	0
$307$	143.760	0.468274	0.234137	−	0.972204i	$-0.424774\pi$
$307$	143.760	0.468274	0.234137	+	0.972204i	$0.424774\pi$
$308$	0	0
$309$	0	0
$310$	769.031	2.48074
$311$	− 101.823i	− 0.327406i	−0.986510	−	0.163703i	$-0.947656\pi$
$311$	− 101.823i	− 0.327406i	0.986510	−	0.163703i	$-0.0523439\pi$
$312$	0	0
$313$	−224.000	−0.715655	−0.357827	−	0.933788i	$-0.616482\pi$
$313$	−224.000	−0.715655	−0.357827	+	0.933788i	$0.616482\pi$
$314$	− 311.085i	− 0.990717i
$315$	0	0
$316$	−259.808	−0.822176
$317$	16.9706i	0.0535349i	0.999642	+	0.0267674i	$0.00852136\pi$
$317$	16.9706i	0.0535349i	−0.999642	+	0.0267674i	$0.991479\pi$
$318$	0	0
$319$	0	0
$320$	− 67.8823i	− 0.212132i
$321$	0	0
$322$	−126.000	−0.391304
$323$	− 42.4264i	− 0.131351i
$324$	0	0
$325$	−162.813	−0.500962
$326$	325.782i	0.999332i
$327$	0	0
$328$	−204.000	−0.621951
$329$	360.075i	1.09445i
$330$	0	0
$331$	−487.000	−1.47130	−0.735650	−	0.677362i	$-0.763123\pi$
$331$	−487.000	−1.47130	−0.735650	+	0.677362i	$0.763123\pi$
$332$	− 191.060i	− 0.575483i
$333$	0	0
$334$	−618.000	−1.85030
$335$	− 347.897i	− 1.03850i
$336$	0	0
$337$	−43.3013	−0.128490	−0.0642452	−	0.997934i	$-0.520464\pi$
$337$	−43.3013	−0.128490	−0.0642452	+	0.997934i	$0.520464\pi$
$338$	384.570i	1.13778i
$339$	0	0
$340$	−415.692	−1.22262
$341$	0	0
$342$	0	0
$343$	594.093	1.73205
$344$	− 220.617i	− 0.641329i
$345$	0	0
$346$	−108.000	−0.312139
$347$	173.914i	0.501192i	0.968092	+	0.250596i	$0.0806266\pi$
$347$	173.914i	0.501192i	−0.968092	+	0.250596i	$0.919373\pi$
$348$	0	0
$349$	632.199	1.81146	0.905729	−	0.423858i	$-0.139325\pi$
$349$	632.199	1.81146	0.905729	+	0.423858i	$0.139325\pi$
$350$	1395.83i	3.98808i
$351$	0	0
$352$	0	0
$353$	− 500.632i	− 1.41822i	−0.705098	−	0.709110i	$-0.749097\pi$
$353$	− 500.632i	− 1.41822i	0.705098	−	0.709110i	$-0.250903\pi$
$354$	0	0
$355$	−648.000	−1.82535
$356$	101.823i	0.286021i
$357$	0	0
$358$	−675.500	−1.88687
$359$	− 205.757i	− 0.573140i	−0.958059	−	0.286570i	$-0.907485\pi$
$359$	− 205.757i	− 0.573140i	0.958059	−	0.286570i	$-0.0925150\pi$
$360$	0	0
$361$	−358.000	−0.991690
$362$	458.055i	1.26534i
$363$	0	0
$364$	−84.0000	−0.230769
$365$	− 514.393i	− 1.40930i
$366$	0	0
$367$	−652.000	−1.77657	−0.888283	−	0.459296i	$-0.848102\pi$
$367$	−652.000	−1.77657	−0.888283	+	0.459296i	$0.848102\pi$
$368$	84.8528i	0.230578i
$369$	0	0
$370$	1267.86	3.42665
$371$	51.4393i	0.138650i
$372$	0	0
$373$	358.535	0.961219	0.480609	−	0.876935i	$-0.340416\pi$
$373$	358.535	0.961219	0.480609	+	0.876935i	$0.340416\pi$
$374$	0	0
$375$	0	0
$376$	145.492	0.386948
$377$	110.309i	0.292596i
$378$	0	0
$379$	158.000	0.416887	0.208443	−	0.978034i	$-0.433160\pi$
$379$	158.000	0.416887	0.208443	+	0.978034i	$0.433160\pi$
$380$	− 29.3939i	− 0.0773523i
$381$	0	0
$382$	−706.677	−1.84994
$383$	267.286i	0.697876i	0.937146	+	0.348938i	$0.113458\pi$
$383$	267.286i	0.697876i	−0.937146	+	0.348938i	$0.886542\pi$
$384$	0	0
$385$	0	0
$386$	− 190.919i	− 0.494608i
$387$	0	0
$388$	−358.000	−0.922680
$389$	377.595i	0.970681i	0.874325	+	0.485341i	$0.161304\pi$
$389$	377.595i	0.970681i	−0.874325	+	0.485341i	$0.838696\pi$
$390$	0	0
$391$	−103.923	−0.265788
$392$	− 480.100i	− 1.22474i
$393$	0	0
$394$	−420.000	−1.06599
$395$	1102.27i	2.79056i
$396$	0	0
$397$	−511.000	−1.28715	−0.643577	−	0.765382i	$-0.722550\pi$
$397$	−511.000	−1.28715	−0.643577	+	0.765382i	$0.722550\pi$
$398$	262.095i	0.658531i
$399$	0	0
$400$	940.000	2.35000
$401$	169.706i	0.423206i	0.977356	+	0.211603i	$0.0678684\pi$
$401$	169.706i	0.423206i	−0.977356	+	0.211603i	$0.932132\pi$
$402$	0	0
$403$	128.172	0.318044
$404$	− 83.2827i	− 0.206145i
$405$	0	0
$406$	945.700	2.32931
$407$	0	0
$408$	0	0
$409$	−29.4449	−0.0719923	−0.0359962	−	0.999352i	$-0.511460\pi$
$409$	−29.4449	−0.0719923	−0.0359962	+	0.999352i	$0.511460\pi$
$410$	− 865.499i	− 2.11097i
$411$	0	0
$412$	290.000	0.703883
$413$	− 257.196i	− 0.622752i
$414$	0	0
$415$	−810.600	−1.95325
$416$	101.823i	0.244768i
$417$	0	0
$418$	0	0
$419$	453.963i	1.08344i	0.840558	+	0.541721i	$0.182227\pi$
$419$	453.963i	1.08344i	−0.840558	+	0.541721i	$0.817773\pi$
$420$	0	0
$421$	−332.000	−0.788599	−0.394299	−	0.918982i	$-0.629013\pi$
$421$	−332.000	−0.788599	−0.394299	+	0.918982i	$0.629013\pi$
$422$	− 224.860i	− 0.532843i
$423$	0	0
$424$	20.7846	0.0490203
$425$	1151.26i	2.70885i
$426$	0	0
$427$	−189.000	−0.442623
$428$	− 93.0806i	− 0.217478i
$429$	0	0
$430$	936.000	2.17674
$431$	311.085i	0.721775i	0.932609	+	0.360888i	$0.117526\pi$
$431$	311.085i	0.721775i	−0.932609	+	0.360888i	$0.882474\pi$
$432$	0	0
$433$	269.000	0.621247	0.310624	−	0.950533i	$-0.399462\pi$
$433$	269.000	0.621247	0.310624	+	0.950533i	$0.399462\pi$
$434$	− 1098.84i	− 2.53190i
$435$	0	0
$436$	−93.5307	−0.214520
$437$	− 7.34847i	− 0.0168157i
$438$	0	0
$439$	−303.109	−0.690453	−0.345227	−	0.938519i	$-0.612198\pi$
$439$	−303.109	−0.690453	−0.345227	+	0.938519i	$0.612198\pi$
$440$	0	0
$441$	0	0
$442$	−207.846	−0.470240
$443$	823.072i	1.85795i	0.370141	+	0.928976i	$0.379309\pi$
$443$	823.072i	1.85795i	−0.370141	+	0.928976i	$0.620691\pi$
$444$	0	0
$445$	432.000	0.970787
$446$	− 173.914i	− 0.389941i
$447$	0	0
$448$	−96.9948	−0.216506
$449$	− 212.132i	− 0.472454i	−0.971698	−	0.236227i	$-0.924089\pi$
$449$	− 212.132i	− 0.472454i	0.971698	−	0.236227i	$-0.0759110\pi$
$450$	0	0
$451$	0	0
$452$	− 76.3675i	− 0.168955i
$453$	0	0
$454$	372.000	0.819383
$455$	356.382i	0.783257i
$456$	0	0
$457$	−128.172	−0.280463	−0.140232	−	0.990119i	$-0.544785\pi$
$457$	−128.172	−0.280463	−0.140232	+	0.990119i	$0.544785\pi$
$458$	195.959i	0.427858i
$459$	0	0
$460$	−72.0000	−0.156522
$461$	7.34847i	0.0159403i	0.999968	+	0.00797014i	$0.00253700\pi$
$461$	7.34847i	0.0159403i	−0.999968	+	0.00797014i	$0.997463\pi$
$462$	0	0
$463$	−436.000	−0.941685	−0.470842	−	0.882217i	$-0.656050\pi$
$463$	−436.000	−0.941685	−0.470842	+	0.882217i	$0.656050\pi$
$464$	− 636.867i	− 1.37256i
$465$	0	0
$466$	−894.000	−1.91845
$467$	882.469i	1.88966i	0.327567	+	0.944828i	$0.393771\pi$
$467$	882.469i	1.88966i	−0.327567	+	0.944828i	$0.606229\pi$
$468$	0	0
$469$	−497.099	−1.05991
$470$	617.271i	1.31334i
$471$	0	0
$472$	−103.923	−0.220176
$473$	0	0
$474$	0	0
$475$	−81.4064	−0.171382
$476$	593.970i	1.24784i
$477$	0	0
$478$	−318.000	−0.665272
$479$	− 379.671i	− 0.792632i	−0.918114	−	0.396316i	$-0.870288\pi$
$479$	− 379.671i	− 0.792632i	0.918114	−	0.396316i	$-0.129712\pi$
$480$	0	0
$481$	211.310	0.439314
$482$	127.279i	0.264065i
$483$	0	0
$484$	0	0
$485$	1518.87i	3.13168i
$486$	0	0
$487$	−704.000	−1.44559	−0.722793	−	0.691065i	$-0.757142\pi$
$487$	−704.000	−1.44559	−0.722793	+	0.691065i	$0.757142\pi$
$488$	76.3675i	0.156491i
$489$	0	0
$490$	2036.89	4.15692
$491$	− 404.166i	− 0.823148i	−0.911376	−	0.411574i	$-0.864979\pi$
$491$	− 404.166i	− 0.823148i	0.911376	−	0.411574i	$-0.135021\pi$
$492$	0	0
$493$	780.000	1.58215
$494$	− 14.6969i	− 0.0297509i
$495$	0	0
$496$	−740.000	−1.49194
$497$	925.907i	1.86299i
$498$	0	0
$499$	523.000	1.04810	0.524048	−	0.851689i	$-0.324421\pi$
$499$	523.000	1.04810	0.524048	+	0.851689i	$0.324421\pi$
$500$	373.352i	0.746705i
$501$	0	0
$502$	−446.869	−0.890178
$503$	117.576i	0.233749i	0.993147	+	0.116874i	$0.0372875\pi$
$503$	117.576i	0.233749i	−0.993147	+	0.116874i	$0.962713\pi$
$504$	0	0
$505$	−353.338	−0.699680
$506$	0	0
$507$	0	0
$508$	342.946	0.675091
$509$	− 496.389i	− 0.975224i	−0.873060	−	0.487612i	$-0.837868\pi$
$509$	− 496.389i	− 0.975224i	0.873060	−	0.487612i	$-0.162132\pi$
$510$	0	0
$511$	−735.000	−1.43836
$512$	− 195.959i	− 0.382733i
$513$	0	0
$514$	1028.84	2.00163
$515$	− 1230.37i	− 2.38906i
$516$	0	0
$517$	0	0
$518$	− 1811.61i	− 3.49731i
$519$	0	0
$520$	144.000	0.276923
$521$	496.389i	0.952762i	0.879239	+	0.476381i	$0.158052\pi$
$521$	496.389i	0.952762i	−0.879239	+	0.476381i	$0.841948\pi$
$522$	0	0
$523$	−462.458	−0.884240	−0.442120	−	0.896956i	$-0.645773\pi$
$523$	−462.458	−0.884240	−0.442120	+	0.896956i	$0.645773\pi$
$524$	− 122.474i	− 0.233730i
$525$	0	0
$526$	330.000	0.627376
$527$	− 906.311i	− 1.71976i
$528$	0	0
$529$	511.000	0.965974
$530$	88.1816i	0.166380i
$531$	0	0
$532$	−42.0000	−0.0789474
$533$	− 144.250i	− 0.270637i
$534$	0	0
$535$	−394.908	−0.738145
$536$	200.858i	0.374735i
$537$	0	0
$538$	−31.1769	−0.0579497
$539$	0	0
$540$	0	0
$541$	197.454	0.364979	0.182490	−	0.983208i	$-0.441584\pi$
$541$	197.454	0.364979	0.182490	+	0.983208i	$0.441584\pi$
$542$	161.220i	0.297455i
$543$	0	0
$544$	720.000	1.32353
$545$	396.817i	0.728105i
$546$	0	0
$547$	−148.956	−0.272315	−0.136158	−	0.990687i	$-0.543475\pi$
$547$	−148.956	−0.272315	−0.136158	+	0.990687i	$0.543475\pi$
$548$	− 254.558i	− 0.464523i
$549$	0	0
$550$	0	0
$551$	55.1543i	0.100099i
$552$	0	0
$553$	1575.00	2.84810
$554$	− 89.0955i	− 0.160822i
$555$	0	0
$556$	−166.277	−0.299059
$557$	− 955.301i	− 1.71508i	−0.514415	−	0.857541i	$-0.671991\pi$
$557$	− 955.301i	− 1.71508i	0.514415	−	0.857541i	$-0.328009\pi$
$558$	0	0
$559$	156.000	0.279070
$560$	− 2057.57i	− 3.67423i
$561$	0	0
$562$	−768.000	−1.36655
$563$	− 524.191i	− 0.931067i	−0.885030	−	0.465534i	$-0.845862\pi$
$563$	− 524.191i	− 0.931067i	0.885030	−	0.465534i	$-0.154138\pi$
$564$	0	0
$565$	−324.000	−0.573451
$566$	852.771i	1.50666i
$567$	0	0
$568$	374.123	0.658667
$569$	156.767i	0.275514i	0.990466	+	0.137757i	$0.0439893\pi$
$569$	156.767i	0.275514i	−0.990466	+	0.137757i	$0.956011\pi$
$570$	0	0
$571$	−999.393	−1.75025	−0.875125	−	0.483896i	$-0.839221\pi$
$571$	−999.393	−1.75025	−0.875125	+	0.483896i	$0.839221\pi$
$572$	0	0
$573$	0	0
$574$	−1236.68	−2.15450
$575$	199.404i	0.346790i
$576$	0	0
$577$	−1013.00	−1.75563	−0.877816	−	0.478997i	$-0.841000\pi$
$577$	−1013.00	−1.75563	−0.877816	+	0.478997i	$0.841000\pi$
$578$	761.791i	1.31798i
$579$	0	0
$580$	540.400	0.931724
$581$	1158.24i	1.99353i
$582$	0	0
$583$	0	0
$584$	296.985i	0.508536i
$585$	0	0
$586$	1032.00	1.76109
$587$	− 526.087i	− 0.896231i	−0.893976	−	0.448115i	$-0.852095\pi$
$587$	− 526.087i	− 0.896231i	0.893976	−	0.448115i	$-0.147905\pi$
$588$	0	0
$589$	64.0859	0.108805
$590$	− 440.908i	− 0.747302i
$591$	0	0
$592$	−1220.00	−2.06081
$593$	560.933i	0.945924i	0.881083	+	0.472962i	$0.156815\pi$
$593$	560.933i	0.945924i	−0.881083	+	0.472962i	$0.843185\pi$
$594$	0	0
$595$	2520.00	4.23529
$596$	39.1918i	0.0657581i
$597$	0	0
$598$	−36.0000	−0.0602007
$599$	− 8.48528i	− 0.0141657i	−0.999975	−	0.00708287i	$-0.997745\pi$
$599$	− 8.48528i	− 0.0141657i	0.999975	−	0.00708287i	$-0.00225457\pi$
$600$	0	0
$601$	−698.016	−1.16143	−0.580713	−	0.814109i	$-0.697226\pi$
$601$	−698.016	−1.16143	−0.580713	+	0.814109i	$0.697226\pi$
$602$	− 1337.42i	− 2.22163i
$603$	0	0
$604$	−6.92820	−0.0114705
$605$	0	0
$606$	0	0
$607$	633.931	1.04437	0.522183	−	0.852833i	$-0.325118\pi$
$607$	633.931	1.04437	0.522183	+	0.852833i	$0.325118\pi$
$608$	50.9117i	0.0837363i
$609$	0	0
$610$	−324.000	−0.531148
$611$	102.879i	0.168377i
$612$	0	0
$613$	−864.293	−1.40994	−0.704970	−	0.709237i	$-0.749040\pi$
$613$	−864.293	−1.40994	−0.704970	+	0.709237i	$0.749040\pi$
$614$	− 352.139i	− 0.573517i
$615$	0	0
$616$	0	0
$617$	848.528i	1.37525i	0.726067	+	0.687624i	$0.241346\pi$
$617$	848.528i	1.37525i	−0.726067	+	0.687624i	$0.758654\pi$
$618$	0	0
$619$	608.000	0.982229	0.491115	−	0.871095i	$-0.336590\pi$
$619$	608.000	0.982229	0.491115	+	0.871095i	$0.336590\pi$
$620$	− 627.911i	− 1.01276i
$621$	0	0
$622$	−249.415	−0.400989
$623$	− 617.271i	− 0.990805i
$624$	0	0
$625$	409.000	0.654400
$626$	548.686i	0.876495i
$627$	0	0
$628$	−254.000	−0.404459
$629$	− 1494.19i	− 2.37550i
$630$	0	0
$631$	436.000	0.690967	0.345483	−	0.938425i	$-0.387715\pi$
$631$	436.000	0.690967	0.345483	+	0.938425i	$0.387715\pi$
$632$	− 636.396i	− 1.00696i
$633$	0	0
$634$	41.5692	0.0655666
$635$	− 1455.00i	− 2.29133i
$636$	0	0
$637$	339.482	0.532939
$638$	0	0
$639$	0	0
$640$	−1163.94	−1.81865
$641$	− 271.529i	− 0.423602i	−0.977313	−	0.211801i	$-0.932067\pi$
$641$	− 271.529i	− 0.423602i	0.977313	−	0.211801i	$-0.0679329\pi$
$642$	0	0
$643$	869.000	1.35148	0.675739	−	0.737141i	$-0.263825\pi$
$643$	869.000	1.35148	0.675739	+	0.737141i	$0.263825\pi$
$644$	102.879i	0.159749i
$645$	0	0
$646$	−103.923	−0.160872
$647$	− 186.676i	− 0.288526i	−0.989539	−	0.144263i	$-0.953919\pi$
$647$	− 186.676i	− 0.288526i	0.989539	−	0.144263i	$-0.0460811\pi$
$648$	0	0
$649$	0	0
$650$	398.808i	0.613551i
$651$	0	0
$652$	266.000	0.407975
$653$	− 1226.12i	− 1.87768i	−0.344358	−	0.938839i	$-0.611903\pi$
$653$	− 1226.12i	− 1.87768i	0.344358	−	0.938839i	$-0.388097\pi$
$654$	0	0
$655$	−519.615	−0.793306
$656$	832.827i	1.26955i
$657$	0	0
$658$	882.000	1.34043
$659$	663.812i	1.00730i	0.863907	+	0.503651i	$0.168010\pi$
$659$	663.812i	1.00730i	−0.863907	+	0.503651i	$0.831990\pi$
$660$	0	0
$661$	−323.000	−0.488654	−0.244327	−	0.969693i	$-0.578567\pi$
$661$	−323.000	−0.488654	−0.244327	+	0.969693i	$0.578567\pi$
$662$	1192.90i	1.80197i
$663$	0	0
$664$	468.000	0.704819
$665$	178.191i	0.267956i
$666$	0	0
$667$	135.100	0.202549
$668$	504.595i	0.755382i
$669$	0	0
$670$	−852.169	−1.27189
$671$	0	0
$672$	0	0
$673$	−587.165	−0.872459	−0.436230	−	0.899835i	$-0.643686\pi$
$673$	−587.165	−0.872459	−0.436230	+	0.899835i	$0.643686\pi$
$674$	106.066i	0.157368i
$675$	0	0
$676$	314.000	0.464497
$677$	416.413i	0.615086i	0.951534	+	0.307543i	$0.0995068\pi$
$677$	416.413i	0.615086i	−0.951534	+	0.307543i	$0.900493\pi$
$678$	0	0
$679$	2170.26	3.19626
$680$	− 1018.23i	− 1.49740i
$681$	0	0
$682$	0	0
$683$	− 852.771i	− 1.24857i	−0.781198	−	0.624283i	$-0.785391\pi$
$683$	− 852.771i	− 1.24857i	0.781198	−	0.624283i	$-0.214609\pi$
$684$	0	0
$685$	−1080.00	−1.57664
$686$	− 1455.23i	− 2.12132i
$687$	0	0
$688$	−900.666	−1.30911
$689$	14.6969i	0.0213308i
$690$	0	0
$691$	−535.000	−0.774240	−0.387120	−	0.922029i	$-0.626530\pi$
$691$	−535.000	−0.774240	−0.387120	+	0.922029i	$0.626530\pi$
$692$	88.1816i	0.127430i
$693$	0	0
$694$	426.000	0.613833
$695$	705.453i	1.01504i
$696$	0	0
$697$	−1020.00	−1.46341
$698$	− 1548.56i	− 2.21857i
$699$	0	0
$700$	1139.69	1.62813
$701$	− 590.327i	− 0.842121i	−0.907032	−	0.421061i	$-0.861658\pi$
$701$	− 590.327i	− 0.842121i	0.907032	−	0.421061i	$-0.138342\pi$
$702$	0	0
$703$	105.655	0.150292
$704$	0	0
$705$	0	0
$706$	−1226.29	−1.73696
$707$	504.874i	0.714108i
$708$	0	0
$709$	−254.000	−0.358251	−0.179126	−	0.983826i	$-0.557327\pi$
$709$	−254.000	−0.358251	−0.179126	+	0.983826i	$0.557327\pi$
$710$	1587.27i	2.23559i
$711$	0	0
$712$	−249.415	−0.350302
$713$	− 156.978i	− 0.220165i
$714$	0	0
$715$	0	0
$716$	551.543i	0.770312i
$717$	0	0
$718$	−504.000	−0.701950
$719$	− 818.830i	− 1.13885i	−0.822045	−	0.569423i	$-0.807167\pi$
$719$	− 818.830i	− 1.13885i	0.822045	−	0.569423i	$-0.192833\pi$
$720$	0	0
$721$	−1758.03	−2.43832
$722$	876.917i	1.21457i
$723$	0	0
$724$	374.000	0.516575
$725$	− 1496.64i	− 2.06433i
$726$	0	0
$727$	514.000	0.707015	0.353508	−	0.935432i	$-0.384989\pi$
$727$	514.000	0.707015	0.353508	+	0.935432i	$0.384989\pi$
$728$	− 205.757i	− 0.282633i
$729$	0	0
$730$	−1260.00	−1.72603
$731$	− 1103.09i	− 1.50901i
$732$	0	0
$733$	−1274.79	−1.73914	−0.869570	−	0.493810i	$-0.835604\pi$
$733$	−1274.79	−1.73914	−0.869570	+	0.493810i	$0.835604\pi$
$734$	1597.07i	2.17584i
$735$	0	0
$736$	124.708	0.169440
$737$	0	0
$738$	0	0
$739$	−791.547	−1.07111	−0.535553	−	0.844502i	$-0.679897\pi$
$739$	−791.547	−1.07111	−0.535553	+	0.844502i	$0.679897\pi$
$740$	− 1035.20i	− 1.39892i
$741$	0	0
$742$	126.000	0.169811
$743$	− 896.513i	− 1.20661i	−0.797510	−	0.603306i	$-0.793850\pi$
$743$	− 896.513i	− 1.20661i	0.797510	−	0.603306i	$-0.206150\pi$
$744$	0	0
$745$	166.277	0.223190
$746$	− 878.227i	− 1.17725i
$747$	0	0
$748$	0	0
$749$	564.271i	0.753366i
$750$	0	0
$751$	−167.000	−0.222370	−0.111185	−	0.993800i	$-0.535465\pi$
$751$	−167.000	−0.222370	−0.111185	+	0.993800i	$0.535465\pi$
$752$	− 593.970i	− 0.789853i
$753$	0	0
$754$	270.200	0.358355
$755$	29.3939i	0.0389323i
$756$	0	0
$757$	−701.000	−0.926024	−0.463012	−	0.886352i	$-0.653231\pi$
$757$	−701.000	−0.926024	−0.463012	+	0.886352i	$0.653231\pi$
$758$	− 387.019i	− 0.510580i
$759$	0	0
$760$	72.0000	0.0947368
$761$	1180.65i	1.55145i	0.631071	+	0.775725i	$0.282616\pi$
$761$	1180.65i	1.55145i	−0.631071	+	0.775725i	$0.717384\pi$
$762$	0	0
$763$	567.000	0.743119
$764$	576.999i	0.755234i
$765$	0	0
$766$	654.715	0.854720
$767$	− 73.4847i	− 0.0958079i
$768$	0	0
$769$	316.965	0.412179	0.206089	−	0.978533i	$-0.433926\pi$
$769$	316.965	0.412179	0.206089	+	0.978533i	$0.433926\pi$
$770$	0	0
$771$	0	0
$772$	−155.885	−0.201923
$773$	− 432.749i	− 0.559831i	−0.960025	−	0.279915i	$-0.909694\pi$
$773$	− 432.749i	− 0.559831i	0.960025	−	0.279915i	$-0.0903065\pi$
$774$	0	0
$775$	−1739.00	−2.24387
$776$	− 876.917i	− 1.13005i
$777$	0	0
$778$	924.915	1.18884
$779$	− 72.1249i	− 0.0925865i
$780$	0	0
$781$	0	0
$782$	254.558i	0.325522i
$783$	0	0
$784$	−1960.00	−2.50000
$785$	1077.63i	1.37278i
$786$	0	0
$787$	855.633	1.08721	0.543604	−	0.839342i	$-0.317059\pi$
$787$	855.633	1.08721	0.543604	+	0.839342i	$0.317059\pi$
$788$	342.929i	0.435189i
$789$	0	0
$790$	2700.00	3.41772
$791$	462.954i	0.585276i
$792$	0	0
$793$	−54.0000	−0.0680958
$794$	1251.69i	1.57643i
$795$	0	0
$796$	214.000	0.268844
$797$	521.845i	0.654761i	0.944893	+	0.327381i	$0.106166\pi$
$797$	521.845i	0.654761i	−0.944893	+	0.327381i	$0.893834\pi$
$798$	0	0
$799$	727.461	0.910465
$800$	− 1381.51i	− 1.72689i
$801$	0	0
$802$	415.692	0.518319
$803$	0	0
$804$	0	0
$805$	436.477	0.542207
$806$	− 313.955i	− 0.389523i
$807$	0	0
$808$	204.000	0.252475
$809$	896.513i	1.10817i	0.832459	+	0.554087i	$0.186933\pi$
$809$	896.513i	1.10817i	−0.832459	+	0.554087i	$0.813067\pi$
$810$	0	0
$811$	−438.209	−0.540332	−0.270166	−	0.962814i	$-0.587079\pi$
$811$	−438.209	−0.540332	−0.270166	+	0.962814i	$0.587079\pi$
$812$	− 772.161i	− 0.950937i
$813$	0	0
$814$	0	0
$815$	− 1128.54i	− 1.38471i
$816$	0	0
$817$	78.0000	0.0954712
$818$	72.1249i	0.0881722i
$819$	0	0
$820$	−706.677	−0.861801
$821$	− 982.245i	− 1.19640i	−0.801346	−	0.598201i	$-0.795883\pi$
$821$	− 982.245i	− 1.19640i	0.801346	−	0.598201i	$-0.204117\pi$
$822$	0	0
$823$	35.0000	0.0425273	0.0212637	−	0.999774i	$-0.493231\pi$
$823$	35.0000	0.0425273	0.0212637	+	0.999774i	$0.493231\pi$
$824$	710.352i	0.862078i
$825$	0	0
$826$	−630.000	−0.762712
$827$	− 1085.12i	− 1.31212i	−0.754708	−	0.656060i	$-0.772222\pi$
$827$	− 1085.12i	− 1.31212i	0.754708	−	0.656060i	$-0.227778\pi$
$828$	0	0
$829$	785.000	0.946924	0.473462	−	0.880814i	$-0.343004\pi$
$829$	785.000	0.946924	0.473462	+	0.880814i	$0.343004\pi$
$830$	1985.56i	2.39224i
$831$	0	0
$832$	−27.7128	−0.0333087
$833$	− 2400.50i	− 2.88175i
$834$	0	0
$835$	2140.81	2.56385
$836$	0	0
$837$	0	0
$838$	1111.98	1.32694
$839$	− 492.146i	− 0.586587i	−0.956022	−	0.293293i	$-0.905249\pi$
$839$	− 492.146i	− 0.586587i	0.956022	−	0.293293i	$-0.0947513\pi$
$840$	0	0
$841$	−173.000	−0.205707
$842$	813.231i	0.965832i
$843$	0	0
$844$	−183.597	−0.217532
$845$	− 1332.19i	− 1.57656i
$846$	0	0
$847$	0	0
$848$	− 84.8528i	− 0.100062i
$849$	0	0
$850$	2820.00	3.31765
$851$	− 258.801i	− 0.304114i
$852$	0	0
$853$	−646.055	−0.757392	−0.378696	−	0.925521i	$-0.623627\pi$
$853$	−646.055	−0.757392	−0.378696	+	0.925521i	$0.623627\pi$
$854$	462.954i	0.542100i
$855$	0	0
$856$	228.000	0.266355
$857$	− 279.242i	− 0.325836i	−0.986640	−	0.162918i	$-0.947909\pi$
$857$	− 279.242i	− 0.325836i	0.986640	−	0.162918i	$-0.0520907\pi$
$858$	0	0
$859$	863.000	1.00466	0.502328	−	0.864677i	$-0.332477\pi$
$859$	863.000	1.00466	0.502328	+	0.864677i	$0.332477\pi$
$860$	− 764.241i	− 0.888652i
$861$	0	0
$862$	762.000	0.883991
$863$	− 173.948i	− 0.201562i	−0.994909	−	0.100781i	$-0.967866\pi$
$863$	− 173.948i	− 0.201562i	0.994909	−	0.100781i	$-0.0321342\pi$
$864$	0	0
$865$	374.123	0.432512
$866$	− 658.913i	− 0.760869i
$867$	0	0
$868$	−897.202	−1.03364
$869$	0	0
$870$	0	0
$871$	−142.028	−0.163063
$872$	− 229.103i	− 0.262732i
$873$	0	0
$874$	−18.0000	−0.0205950
$875$	− 2263.33i	− 2.58666i
$876$	0	0
$877$	−850.437	−0.969711	−0.484856	−	0.874594i	$-0.661128\pi$
$877$	−850.437	−0.969711	−0.484856	+	0.874594i	$0.661128\pi$
$878$	742.462i	0.845629i
$879$	0	0
$880$	0	0
$881$	− 131.522i	− 0.149287i	−0.997210	−	0.0746435i	$-0.976218\pi$
$881$	− 131.522i	− 0.149287i	0.997210	−	0.0746435i	$-0.0237819\pi$
$882$	0	0
$883$	1561.00	1.76784	0.883918	−	0.467641i	$-0.154896\pi$
$883$	1561.00	1.76784	0.883918	+	0.467641i	$0.154896\pi$
$884$	169.706i	0.191975i
$885$	0	0
$886$	2016.11	2.27552
$887$	− 1139.01i	− 1.28412i	−0.766655	−	0.642059i	$-0.778080\pi$
$887$	− 1139.01i	− 1.28412i	0.766655	−	0.642059i	$-0.221920\pi$
$888$	0	0
$889$	−2079.00	−2.33858
$890$	− 1058.18i	− 1.18897i
$891$	0	0
$892$	−142.000	−0.159193
$893$	51.4393i	0.0576028i
$894$	0	0
$895$	2340.00	2.61453
$896$	1663.12i	1.85616i
$897$	0	0
$898$	−519.615	−0.578636
$899$	1178.20i	1.31057i
$900$	0	0
$901$	103.923	0.115342
$902$	0	0
$903$	0	0
$904$	187.061	0.206926
$905$	− 1586.75i	− 1.75331i
$906$	0	0
$907$	533.000	0.587652	0.293826	−	0.955859i	$-0.405071\pi$
$907$	533.000	0.587652	0.293826	+	0.955859i	$0.405071\pi$
$908$	− 303.737i	− 0.334512i
$909$	0	0
$910$	872.954	0.959290
$911$	1094.60i	1.20154i	0.799423	+	0.600769i	$0.205139\pi$
$911$	1094.60i	1.20154i	−0.799423	+	0.600769i	$0.794861\pi$
$912$	0	0
$913$	0	0
$914$	313.955i	0.343496i
$915$	0	0
$916$	160.000	0.174672
$917$	742.462i	0.809664i
$918$	0	0
$919$	1633.32	1.77728	0.888642	−	0.458602i	$-0.151650\pi$
$919$	1633.32	1.77728	0.888642	+	0.458602i	$0.151650\pi$
$920$	− 176.363i	− 0.191699i
$921$	0	0
$922$	18.0000	0.0195228
$923$	264.545i	0.286614i
$924$	0	0
$925$	−2867.00	−3.09946
$926$	1067.98i	1.15332i
$927$	0	0
$928$	−936.000	−1.00862
$929$	1022.48i	1.10062i	0.834960	+	0.550310i	$0.185491\pi$
$929$	1022.48i	1.10062i	−0.834960	+	0.550310i	$0.814509\pi$
$930$	0	0
$931$	169.741	0.182321
$932$	729.948i	0.783206i
$933$	0	0
$934$	2161.60	2.31435
$935$	0	0
$936$	0	0
$937$	−136.832	−0.146032	−0.0730160	−	0.997331i	$-0.523262\pi$
$937$	−136.832	−0.146032	−0.0730160	+	0.997331i	$0.523262\pi$
$938$	1217.64i	1.29812i
$939$	0	0
$940$	504.000	0.536170
$941$	1354.57i	1.43950i	0.694234	+	0.719749i	$0.255743\pi$
$941$	1354.57i	1.43950i	−0.694234	+	0.719749i	$0.744257\pi$
$942$	0	0
$943$	−176.669	−0.187348
$944$	424.264i	0.449432i
$945$	0	0
$946$	0	0
$947$	− 1582.50i	− 1.67107i	−0.549436	−	0.835536i	$-0.685157\pi$
$947$	− 1582.50i	− 1.67107i	0.549436	−	0.835536i	$-0.314843\pi$
$948$	0	0
$949$	−210.000	−0.221286
$950$	199.404i	0.209899i
$951$	0	0
$952$	−1454.92	−1.52828
$953$	1006.74i	1.05639i	0.849123	+	0.528195i	$0.177131\pi$
$953$	1006.74i	1.05639i	−0.849123	+	0.528195i	$0.822869\pi$
$954$	0	0
$955$	2448.00	2.56335
$956$	259.646i	0.271596i
$957$	0	0
$958$	−930.000	−0.970772
$959$	1543.18i	1.60915i
$960$	0	0
$961$	408.000	0.424558
$962$	− 517.602i	− 0.538048i
$963$	0	0
$964$	103.923	0.107804
$965$	661.362i	0.685349i
$966$	0	0
$967$	−1491.30	−1.54219	−0.771094	−	0.636721i	$-0.780290\pi$
$967$	−1491.30	−1.54219	−0.771094	+	0.636721i	$0.780290\pi$
$968$	0	0
$969$	0	0
$970$	3720.45	3.83551
$971$	886.712i	0.913195i	0.889673	+	0.456597i	$0.150932\pi$
$971$	886.712i	0.913195i	−0.889673	+	0.456597i	$0.849068\pi$
$972$	0	0
$973$	1008.00	1.03597
$974$	1724.44i	1.77047i
$975$	0	0
$976$	311.769	0.319436
$977$	− 190.919i	− 0.195413i	−0.995215	−	0.0977067i	$-0.968849\pi$
$977$	− 190.919i	− 0.195413i	0.995215	−	0.0977067i	$-0.0311507\pi$
$978$	0	0
$979$	0	0
$980$	− 1663.12i	− 1.69706i
$981$	0	0
$982$	−990.000	−1.00815
$983$	895.197i	0.910679i	0.890318	+	0.455339i	$0.150482\pi$
$983$	895.197i	0.910679i	−0.890318	+	0.455339i	$0.849518\pi$
$984$	0	0
$985$	1454.92	1.47708
$986$	− 1910.60i	− 1.93773i
$987$	0	0
$988$	−12.0000	−0.0121457
$989$	− 191.060i	− 0.193185i
$990$	0	0
$991$	−1334.00	−1.34612	−0.673058	−	0.739590i	$-0.735019\pi$
$991$	−1334.00	−1.34612	−0.673058	+	0.739590i	$0.735019\pi$
$992$	1087.57i	1.09634i
$993$	0	0
$994$	2268.00	2.28169
$995$	− 907.925i	− 0.912488i
$996$	0	0
$997$	995.929	0.998926	0.499463	−	0.866335i	$-0.333531\pi$
$997$	995.929	0.998926	0.499463	+	0.866335i	$0.333531\pi$
$998$	− 1281.08i	− 1.28365i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.e.485.2	yes	4
3.2	odd	2	inner	1089.3.b.e.485.3	yes	4
11.10	odd	2	inner	1089.3.b.e.485.4	yes	4
33.32	even	2	inner	1089.3.b.e.485.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.b.e.485.1	✓	4	33.32	even	2	inner
1089.3.b.e.485.2	yes	4	1.1	even	1	trivial
1089.3.b.e.485.3	yes	4	3.2	odd	2	inner
1089.3.b.e.485.4	yes	4	11.10	odd	2	inner

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.44949i q^{2} -2.00000 q^{4} +8.48528i q^{5} +12.1244 q^{7} -4.89898i q^{8} +O(q^{10})\)	\(q-2.44949i q^{2} -2.00000 q^{4} +8.48528i q^{5} +12.1244 q^{7} -4.89898i q^{8} +20.7846 q^{10} +3.46410 q^{13} -29.6985i q^{14} -20.0000 q^{16} -24.4949i q^{17} +1.73205 q^{19} -16.9706i q^{20} -4.24264i q^{23} -47.0000 q^{25} -8.48528i q^{26} -24.2487 q^{28} +31.8434i q^{29} +37.0000 q^{31} +29.3939i q^{32} -60.0000 q^{34} +102.879i q^{35} +61.0000 q^{37} -4.24264i q^{38} +41.5692 q^{40} -41.6413i q^{41} +45.0333 q^{43} -10.3923 q^{46} +29.6985i q^{47} +98.0000 q^{49} +115.126i q^{50} -6.92820 q^{52} +4.24264i q^{53} -59.3970i q^{56} +78.0000 q^{58} -21.2132i q^{59} -15.5885 q^{61} -90.6311i q^{62} -8.00000 q^{64} +29.3939i q^{65} -41.0000 q^{67} +48.9898i q^{68} +252.000 q^{70} +76.3675i q^{71} -60.6218 q^{73} -149.419i q^{74} -3.46410 q^{76} +129.904 q^{79} -169.706i q^{80} -102.000 q^{82} +95.5301i q^{83} +207.846 q^{85} -110.309i q^{86} -50.9117i q^{89} +42.0000 q^{91} +8.48528i q^{92} +72.7461 q^{94} +14.6969i q^{95} +179.000 q^{97} -240.050i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} - 80 q^{16} - 188 q^{25} + 148 q^{31} - 240 q^{34} + 244 q^{37} + 392 q^{49} + 312 q^{58} - 32 q^{64} - 164 q^{67} + 1008 q^{70} - 408 q^{82} + 168 q^{91} + 716 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 - 80 * q^16 - 188 * q^25 + 148 * q^31 - 240 * q^34 + 244 * q^37 + 392 * q^49 + 312 * q^58 - 32 * q^64 - 164 * q^67 + 1008 * q^70 - 408 * q^82 + 168 * q^91 + 716 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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