LMFDB - Embedded newform 189.3.b.c.134.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,3,Mod(134,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, 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("189.134");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	189.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:	$5.14987699641$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			134.4
Root			$-1.27733i$ of defining polynomial
Character	$\chi$	$=$	189.134
Dual form			189.3.b.c.134.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.722666i q^{2} +3.47775 q^{4} -7.94527i q^{5} +2.64575 q^{7} -5.40392i q^{8} +O(q^{10})$	\(q-0.722666i q^{2} +3.47775 q^{4} -7.94527i q^{5} +2.64575 q^{7} -5.40392i q^{8} -5.74177 q^{10} +9.25195i q^{11} -22.1456 q^{13} -1.91199i q^{14} +10.0058 q^{16} -27.2692i q^{17} +31.8633 q^{19} -27.6317i q^{20} +6.68607 q^{22} +13.5294i q^{23} -38.1273 q^{25} +16.0038i q^{26} +9.20127 q^{28} -6.73870i q^{29} +11.9351 q^{31} -28.8465i q^{32} -19.7065 q^{34} -21.0212i q^{35} +26.8988 q^{37} -23.0265i q^{38} -42.9356 q^{40} +63.9635i q^{41} -16.2185 q^{43} +32.1760i q^{44} +9.77724 q^{46} +0.518396i q^{47} +7.00000 q^{49} +27.5533i q^{50} -77.0168 q^{52} +71.8912i q^{53} +73.5092 q^{55} -14.2974i q^{56} -4.86983 q^{58} +38.5540i q^{59} +27.9222 q^{61} -8.62506i q^{62} +19.1768 q^{64} +175.953i q^{65} +29.5649 q^{67} -94.8356i q^{68} -15.1913 q^{70} +39.4892i q^{71} +67.1187 q^{73} -19.4388i q^{74} +110.813 q^{76} +24.4784i q^{77} -18.7689 q^{79} -79.4987i q^{80} +46.2242 q^{82} +80.0893i q^{83} -216.661 q^{85} +11.7205i q^{86} +49.9968 q^{88} -62.9293i q^{89} -58.5917 q^{91} +47.0520i q^{92} +0.374627 q^{94} -253.162i q^{95} -46.7811 q^{97} -5.05866i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{4}+O(q^{10}) $ 8 * q - 12 * q^4	$ 8 q - 12 q^{4} - 52 q^{10} + 36 q^{13} + 132 q^{16} + 12 q^{19} - 136 q^{22} - 108 q^{25} + 56 q^{28} - 28 q^{31} - 12 q^{34} - 4 q^{37} + 336 q^{40} - 152 q^{43} + 108 q^{46} + 56 q^{49} - 272 q^{52} + 196 q^{55} - 220 q^{58} + 180 q^{61} - 700 q^{64} - 132 q^{67} + 196 q^{70} + 272 q^{73} + 544 q^{76} + 316 q^{79} + 28 q^{82} - 228 q^{85} - 56 q^{91} - 348 q^{94} - 364 q^{97}+O(q^{100}) $ 8 * q - 12 * q^4 - 52 * q^10 + 36 * q^13 + 132 * q^16 + 12 * q^19 - 136 * q^22 - 108 * q^25 + 56 * q^28 - 28 * q^31 - 12 * q^34 - 4 * q^37 + 336 * q^40 - 152 * q^43 + 108 * q^46 + 56 * q^49 - 272 * q^52 + 196 * q^55 - 220 * q^58 + 180 * q^61 - 700 * q^64 - 132 * q^67 + 196 * q^70 + 272 * q^73 + 544 * q^76 + 316 * q^79 + 28 * q^82 - 228 * q^85 - 56 * q^91 - 348 * q^94 - 364 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 0.722666i	− 0.361333i	−0.983544	−	0.180666i	$-0.942175\pi$
$2$	− 0.722666i	− 0.361333i	0.983544	−	0.180666i	$-0.0578254\pi$
$3$	0	0
$3$	0	0
$4$	3.47775	0.869439
$5$	− 7.94527i	− 1.58905i	−0.607229	−	0.794527i	$-0.707719\pi$
$5$	− 7.94527i	− 1.58905i	0.607229	−	0.794527i	$-0.292281\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	− 5.40392i	− 0.675490i
$9$	0	0
$10$	−5.74177	−0.574177
$11$	9.25195i	0.841086i	0.907273	+	0.420543i	$0.138160\pi$
$11$	9.25195i	0.841086i	−0.907273	+	0.420543i	$0.861840\pi$
$12$	0	0
$13$	−22.1456	−1.70351	−0.851753	−	0.523944i	$-0.824460\pi$
$13$	−22.1456	−1.70351	−0.851753	+	0.523944i	$0.824460\pi$
$14$	− 1.91199i	− 0.136571i
$15$	0	0
$16$	10.0058	0.625362
$17$	− 27.2692i	− 1.60407i	−0.597277	−	0.802035i	$-0.703751\pi$
$17$	− 27.2692i	− 1.60407i	0.597277	−	0.802035i	$-0.296249\pi$
$18$	0	0
$19$	31.8633	1.67702	0.838508	−	0.544890i	$-0.183428\pi$
$19$	31.8633	1.67702	0.838508	+	0.544890i	$0.183428\pi$
$20$	− 27.6317i	− 1.38158i
$21$	0	0
$22$	6.68607	0.303912
$23$	13.5294i	0.588235i	0.955769	+	0.294118i	$0.0950257\pi$
$23$	13.5294i	0.588235i	−0.955769	+	0.294118i	$0.904974\pi$
$24$	0	0
$25$	−38.1273	−1.52509
$26$	16.0038i	0.615532i
$27$	0	0
$28$	9.20127	0.328617
$29$	− 6.73870i	− 0.232369i	−0.993228	−	0.116184i	$-0.962934\pi$
$29$	− 6.73870i	− 0.232369i	0.993228	−	0.116184i	$-0.0370664\pi$
$30$	0	0
$31$	11.9351	0.385002	0.192501	−	0.981297i	$-0.438340\pi$
$31$	11.9351	0.385002	0.192501	+	0.981297i	$0.438340\pi$
$32$	− 28.8465i	− 0.901453i
$33$	0	0
$34$	−19.7065	−0.579603
$35$	− 21.0212i	− 0.600606i
$36$	0	0
$37$	26.8988	0.726993	0.363497	−	0.931595i	$-0.381583\pi$
$37$	26.8988	0.726993	0.363497	+	0.931595i	$0.381583\pi$
$38$	− 23.0265i	− 0.605961i
$39$	0	0
$40$	−42.9356	−1.07339
$41$	63.9635i	1.56008i	0.625727	+	0.780042i	$0.284803\pi$
$41$	63.9635i	1.56008i	−0.625727	+	0.780042i	$0.715197\pi$
$42$	0	0
$43$	−16.2185	−0.377174	−0.188587	−	0.982056i	$-0.560391\pi$
$43$	−16.2185	−0.377174	−0.188587	+	0.982056i	$0.560391\pi$
$44$	32.1760i	0.731273i
$45$	0	0
$46$	9.77724	0.212549
$47$	0.518396i	0.0110297i	0.999985	+	0.00551485i	$0.00175544\pi$
$47$	0.518396i	0.0110297i	−0.999985	+	0.00551485i	$0.998245\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	27.5533i	0.551066i
$51$	0	0
$52$	−77.0168	−1.48109
$53$	71.8912i	1.35644i	0.734860	+	0.678219i	$0.237248\pi$
$53$	71.8912i	1.35644i	−0.734860	+	0.678219i	$0.762752\pi$
$54$	0	0
$55$	73.5092	1.33653
$56$	− 14.2974i	− 0.255311i
$57$	0	0
$58$	−4.86983	−0.0839625
$59$	38.5540i	0.653457i	0.945118	+	0.326729i	$0.105946\pi$
$59$	38.5540i	0.653457i	−0.945118	+	0.326729i	$0.894054\pi$
$60$	0	0
$61$	27.9222	0.457742	0.228871	−	0.973457i	$-0.426497\pi$
$61$	27.9222	0.457742	0.228871	+	0.973457i	$0.426497\pi$
$62$	− 8.62506i	− 0.139114i
$63$	0	0
$64$	19.1768	0.299637
$65$	175.953i	2.70696i
$66$	0	0
$67$	29.5649	0.441266	0.220633	−	0.975357i	$-0.429188\pi$
$67$	29.5649	0.441266	0.220633	+	0.975357i	$0.429188\pi$
$68$	− 94.8356i	− 1.39464i
$69$	0	0
$70$	−15.1913	−0.217019
$71$	39.4892i	0.556186i	0.960554	+	0.278093i	$0.0897023\pi$
$71$	39.4892i	0.556186i	−0.960554	+	0.278093i	$0.910298\pi$
$72$	0	0
$73$	67.1187	0.919434	0.459717	−	0.888066i	$-0.347951\pi$
$73$	67.1187	0.919434	0.459717	+	0.888066i	$0.347951\pi$
$74$	− 19.4388i	− 0.262687i
$75$	0	0
$76$	110.813	1.45806
$77$	24.4784i	0.317901i
$78$	0	0
$79$	−18.7689	−0.237581	−0.118790	−	0.992919i	$-0.537902\pi$
$79$	−18.7689	−0.237581	−0.118790	+	0.992919i	$0.537902\pi$
$80$	− 79.4987i	− 0.993734i
$81$	0	0
$82$	46.2242	0.563710
$83$	80.0893i	0.964931i	0.875915	+	0.482466i	$0.160259\pi$
$83$	80.0893i	0.964931i	−0.875915	+	0.482466i	$0.839741\pi$
$84$	0	0
$85$	−216.661	−2.54895
$86$	11.7205i	0.136285i
$87$	0	0
$88$	49.9968	0.568145
$89$	− 62.9293i	− 0.707071i	−0.935421	−	0.353535i	$-0.884979\pi$
$89$	− 62.9293i	− 0.707071i	0.935421	−	0.353535i	$-0.115021\pi$
$90$	0	0
$91$	−58.5917	−0.643864
$92$	47.0520i	0.511434i
$93$	0	0
$94$	0.374627	0.00398539
$95$	− 253.162i	− 2.66487i
$96$	0	0
$97$	−46.7811	−0.482280	−0.241140	−	0.970490i	$-0.577521\pi$
$97$	−46.7811	−0.482280	−0.241140	+	0.970490i	$0.577521\pi$
$98$	− 5.05866i	− 0.0516190i
$99$	0	0
$100$	−132.597	−1.32597
$101$	− 104.056i	− 1.03026i	−0.857113	−	0.515128i	$-0.827745\pi$
$101$	− 104.056i	− 1.03026i	0.857113	−	0.515128i	$-0.172255\pi$
$102$	0	0
$103$	−103.860	−1.00835	−0.504174	−	0.863602i	$-0.668203\pi$
$103$	−103.860	−1.00835	−0.504174	+	0.863602i	$0.668203\pi$
$104$	119.673i	1.15070i
$105$	0	0
$106$	51.9533	0.490126
$107$	116.531i	1.08907i	0.838738	+	0.544536i	$0.183294\pi$
$107$	116.531i	1.08907i	−0.838738	+	0.544536i	$0.816706\pi$
$108$	0	0
$109$	−13.9837	−0.128291	−0.0641456	−	0.997941i	$-0.520432\pi$
$109$	−13.9837	−0.128291	−0.0641456	+	0.997941i	$0.520432\pi$
$110$	− 53.1226i	− 0.482933i
$111$	0	0
$112$	26.4728	0.236365
$113$	52.5714i	0.465233i	0.972569	+	0.232617i	$0.0747287\pi$
$113$	52.5714i	0.465233i	−0.972569	+	0.232617i	$0.925271\pi$
$114$	0	0
$115$	107.495	0.934737
$116$	− 23.4355i	− 0.202030i
$117$	0	0
$118$	27.8616	0.236116
$119$	− 72.1475i	− 0.606282i
$120$	0	0
$121$	35.4014	0.292574
$122$	− 20.1784i	− 0.165397i
$123$	0	0
$124$	41.5072	0.334736
$125$	104.300i	0.834400i
$126$	0	0
$127$	23.1204	0.182051	0.0910254	−	0.995849i	$-0.470986\pi$
$127$	23.1204	0.182051	0.0910254	+	0.995849i	$0.470986\pi$
$128$	− 129.244i	− 1.00972i
$129$	0	0
$130$	127.155	0.978114
$131$	− 52.3026i	− 0.399256i	−0.979872	−	0.199628i	$-0.936027\pi$
$131$	− 52.3026i	− 0.399256i	0.979872	−	0.199628i	$-0.0639734\pi$
$132$	0	0
$133$	84.3023	0.633852
$134$	− 21.3655i	− 0.159444i
$135$	0	0
$136$	−147.360	−1.08353
$137$	− 145.292i	− 1.06052i	−0.847834	−	0.530261i	$-0.822094\pi$
$137$	− 145.292i	− 1.06052i	0.847834	−	0.530261i	$-0.177906\pi$
$138$	0	0
$139$	48.7826	0.350954	0.175477	−	0.984484i	$-0.443853\pi$
$139$	48.7826	0.350954	0.175477	+	0.984484i	$0.443853\pi$
$140$	− 73.1066i	− 0.522190i
$141$	0	0
$142$	28.5375	0.200968
$143$	− 204.890i	− 1.43280i
$144$	0	0
$145$	−53.5408	−0.369247
$146$	− 48.5043i	− 0.332222i
$147$	0	0
$148$	93.5473	0.632076
$149$	221.450i	1.48624i	0.669157	+	0.743121i	$0.266655\pi$
$149$	221.450i	1.48624i	−0.669157	+	0.743121i	$0.733345\pi$
$150$	0	0
$151$	−282.901	−1.87352	−0.936759	−	0.349975i	$-0.886190\pi$
$151$	−282.901	−1.87352	−0.936759	+	0.349975i	$0.886190\pi$
$152$	− 172.187i	− 1.13281i
$153$	0	0
$154$	17.6897	0.114868
$155$	− 94.8273i	− 0.611789i
$156$	0	0
$157$	34.3984	0.219098	0.109549	−	0.993981i	$-0.465059\pi$
$157$	34.3984	0.219098	0.109549	+	0.993981i	$0.465059\pi$
$158$	13.5636i	0.0858458i
$159$	0	0
$160$	−229.193	−1.43246
$161$	35.7954i	0.222332i
$162$	0	0
$163$	−0.514328	−0.00315539	−0.00157769	−	0.999999i	$-0.500502\pi$
$163$	−0.514328	−0.00315539	−0.00157769	+	0.999999i	$0.500502\pi$
$164$	222.449i	1.35640i
$165$	0	0
$166$	57.8778	0.348661
$167$	− 156.191i	− 0.935275i	−0.883920	−	0.467638i	$-0.845105\pi$
$167$	− 156.191i	− 0.935275i	0.883920	−	0.467638i	$-0.154895\pi$
$168$	0	0
$169$	321.426	1.90193
$170$	156.574i	0.921021i
$171$	0	0
$172$	−56.4039	−0.327930
$173$	− 76.3720i	− 0.441457i	−0.975335	−	0.220728i	$-0.929157\pi$
$173$	− 76.3720i	− 0.441457i	0.975335	−	0.220728i	$-0.0708434\pi$
$174$	0	0
$175$	−100.875	−0.576431
$176$	92.5731i	0.525984i
$177$	0	0
$178$	−45.4768	−0.255488
$179$	− 92.6544i	− 0.517622i	−0.965928	−	0.258811i	$-0.916669\pi$
$179$	− 92.6544i	− 0.517622i	0.965928	−	0.258811i	$-0.0833307\pi$
$180$	0	0
$181$	−150.103	−0.829296	−0.414648	−	0.909982i	$-0.636095\pi$
$181$	−150.103	−0.829296	−0.414648	+	0.909982i	$0.636095\pi$
$182$	42.3422i	0.232649i
$183$	0	0
$184$	73.1118	0.397347
$185$	− 213.718i	− 1.15523i
$186$	0	0
$187$	252.293	1.34916
$188$	1.80285i	0.00958964i
$189$	0	0
$190$	−182.952	−0.962904
$191$	226.023i	1.18337i	0.806171	+	0.591683i	$0.201536\pi$
$191$	226.023i	1.18337i	−0.806171	+	0.591683i	$0.798464\pi$
$192$	0	0
$193$	35.4187	0.183517	0.0917583	−	0.995781i	$-0.470751\pi$
$193$	35.4187	0.183517	0.0917583	+	0.995781i	$0.470751\pi$
$194$	33.8071i	0.174264i
$195$	0	0
$196$	24.3443	0.124206
$197$	93.8848i	0.476572i	0.971195	+	0.238286i	$0.0765856\pi$
$197$	93.8848i	0.476572i	−0.971195	+	0.238286i	$0.923414\pi$
$198$	0	0
$199$	−20.0037	−0.100521	−0.0502607	−	0.998736i	$-0.516005\pi$
$199$	−20.0037	−0.100521	−0.0502607	+	0.998736i	$0.516005\pi$
$200$	206.037i	1.03018i
$201$	0	0
$202$	−75.1975	−0.372265
$203$	− 17.8289i	− 0.0878272i
$204$	0	0
$205$	508.207	2.47906
$206$	75.0559i	0.364349i
$207$	0	0
$208$	−221.584	−1.06531
$209$	294.798i	1.41051i
$210$	0	0
$211$	241.002	1.14219	0.571095	−	0.820884i	$-0.306519\pi$
$211$	241.002	1.14219	0.571095	+	0.820884i	$0.306519\pi$
$212$	250.020i	1.17934i
$213$	0	0
$214$	84.2127	0.393517
$215$	128.860i	0.599350i
$216$	0	0
$217$	31.5772	0.145517
$218$	10.1056i	0.0463558i
$219$	0	0
$220$	255.647	1.16203
$221$	603.892i	2.73254i
$222$	0	0
$223$	−358.666	−1.60837	−0.804185	−	0.594380i	$-0.797398\pi$
$223$	−358.666	−1.60837	−0.804185	+	0.594380i	$0.797398\pi$
$224$	− 76.3207i	− 0.340717i
$225$	0	0
$226$	37.9915	0.168104
$227$	− 126.918i	− 0.559111i	−0.960129	−	0.279556i	$-0.909813\pi$
$227$	− 126.918i	− 0.559111i	0.960129	−	0.279556i	$-0.0901872\pi$
$228$	0	0
$229$	−240.003	−1.04805	−0.524024	−	0.851703i	$-0.675570\pi$
$229$	−240.003	−1.04805	−0.524024	+	0.851703i	$0.675570\pi$
$230$	− 77.6828i	− 0.337751i
$231$	0	0
$232$	−36.4154	−0.156963
$233$	− 353.670i	− 1.51790i	−0.651150	−	0.758949i	$-0.725713\pi$
$233$	− 353.670i	− 1.51790i	0.651150	−	0.758949i	$-0.274287\pi$
$234$	0	0
$235$	4.11879	0.0175268
$236$	134.081i	0.568141i
$237$	0	0
$238$	−52.1385	−0.219069
$239$	− 460.988i	− 1.92882i	−0.264413	−	0.964410i	$-0.585178\pi$
$239$	− 460.988i	− 1.92882i	0.264413	−	0.964410i	$-0.414822\pi$
$240$	0	0
$241$	−286.281	−1.18789	−0.593944	−	0.804507i	$-0.702430\pi$
$241$	−286.281	−1.18789	−0.593944	+	0.804507i	$0.702430\pi$
$242$	− 25.5834i	− 0.105716i
$243$	0	0
$244$	97.1067	0.397978
$245$	− 55.6169i	− 0.227008i
$246$	0	0
$247$	−705.631	−2.85680
$248$	− 64.4961i	− 0.260065i
$249$	0	0
$250$	75.3741	0.301496
$251$	171.893i	0.684834i	0.939548	+	0.342417i	$0.111246\pi$
$251$	171.893i	0.684834i	−0.939548	+	0.342417i	$0.888754\pi$
$252$	0	0
$253$	−125.173	−0.494757
$254$	− 16.7083i	− 0.0657809i
$255$	0	0
$256$	−16.6934	−0.0652084
$257$	477.109i	1.85645i	0.372015	+	0.928227i	$0.378667\pi$
$257$	477.109i	1.85645i	−0.372015	+	0.928227i	$0.621333\pi$
$258$	0	0
$259$	71.1674	0.274778
$260$	611.920i	2.35354i
$261$	0	0
$262$	−37.7973	−0.144264
$263$	− 309.722i	− 1.17765i	−0.808260	−	0.588826i	$-0.799590\pi$
$263$	− 309.722i	− 1.17765i	0.808260	−	0.588826i	$-0.200410\pi$
$264$	0	0
$265$	571.195	2.15545
$266$	− 60.9224i	− 0.229032i
$267$	0	0
$268$	102.819	0.383654
$269$	2.34739i	0.00872637i	0.999990	+	0.00436318i	$0.00138885\pi$
$269$	2.34739i	0.00872637i	−0.999990	+	0.00436318i	$0.998611\pi$
$270$	0	0
$271$	13.0921	0.0483105	0.0241553	−	0.999708i	$-0.492310\pi$
$271$	13.0921	0.0483105	0.0241553	+	0.999708i	$0.492310\pi$
$272$	− 272.850i	− 1.00312i
$273$	0	0
$274$	−104.997	−0.383202
$275$	− 352.752i	− 1.28273i
$276$	0	0
$277$	223.245	0.805937	0.402968	−	0.915214i	$-0.367978\pi$
$277$	223.245	0.805937	0.402968	+	0.915214i	$0.367978\pi$
$278$	− 35.2535i	− 0.126811i
$279$	0	0
$280$	−113.597	−0.405703
$281$	− 42.6462i	− 0.151766i	−0.997117	−	0.0758830i	$-0.975822\pi$
$281$	− 42.6462i	− 0.151766i	0.997117	−	0.0758830i	$-0.0241775\pi$
$282$	0	0
$283$	−103.687	−0.366386	−0.183193	−	0.983077i	$-0.558643\pi$
$283$	−103.687	−0.366386	−0.183193	+	0.983077i	$0.558643\pi$
$284$	137.334i	0.483569i
$285$	0	0
$286$	−148.067	−0.517716
$287$	169.231i	0.589657i
$288$	0	0
$289$	−454.609	−1.57304
$290$	38.6921i	0.133421i
$291$	0	0
$292$	233.422	0.799391
$293$	− 322.920i	− 1.10212i	−0.834467	−	0.551058i	$-0.814224\pi$
$293$	− 322.920i	− 1.10212i	0.834467	−	0.551058i	$-0.185776\pi$
$294$	0	0
$295$	306.322	1.03838
$296$	− 145.359i	− 0.491076i
$297$	0	0
$298$	160.034	0.537028
$299$	− 299.616i	− 1.00206i
$300$	0	0
$301$	−42.9101	−0.142558
$302$	204.443i	0.676964i
$303$	0	0
$304$	318.818	1.04874
$305$	− 221.850i	− 0.727376i
$306$	0	0
$307$	553.767	1.80380	0.901901	−	0.431943i	$-0.142172\pi$
$307$	553.767	1.80380	0.901901	+	0.431943i	$0.142172\pi$
$308$	85.1297i	0.276395i
$309$	0	0
$310$	−68.5284	−0.221059
$311$	281.881i	0.906370i	0.891416	+	0.453185i	$0.149712\pi$
$311$	281.881i	0.906370i	−0.891416	+	0.453185i	$0.850288\pi$
$312$	0	0
$313$	−423.175	−1.35200	−0.675999	−	0.736903i	$-0.736288\pi$
$313$	−423.175	−1.35200	−0.675999	+	0.736903i	$0.736288\pi$
$314$	− 24.8586i	− 0.0791674i
$315$	0	0
$316$	−65.2736	−0.206562
$317$	163.024i	0.514272i	0.966375	+	0.257136i	$0.0827788\pi$
$317$	163.024i	0.514272i	−0.966375	+	0.257136i	$0.917221\pi$
$318$	0	0
$319$	62.3461	0.195442
$320$	− 152.365i	− 0.476140i
$321$	0	0
$322$	25.8681	0.0803358
$323$	− 868.886i	− 2.69005i
$324$	0	0
$325$	844.351	2.59800
$326$	0.371687i	0.00114015i
$327$	0	0
$328$	345.653	1.05382
$329$	1.37155i	0.00416883i
$330$	0	0
$331$	5.69024	0.0171911	0.00859553	−	0.999963i	$-0.497264\pi$
$331$	5.69024	0.0171911	0.00859553	+	0.999963i	$0.497264\pi$
$332$	278.531i	0.838948i
$333$	0	0
$334$	−112.874	−0.337946
$335$	− 234.901i	− 0.701196i
$336$	0	0
$337$	−642.277	−1.90587	−0.952933	−	0.303182i	$-0.901951\pi$
$337$	−642.277	−1.90587	−0.952933	+	0.303182i	$0.901951\pi$
$338$	− 232.284i	− 0.687230i
$339$	0	0
$340$	−753.494	−2.21616
$341$	110.423i	0.323820i
$342$	0	0
$343$	18.5203	0.0539949
$344$	87.6433i	0.254777i
$345$	0	0
$346$	−55.1914	−0.159513
$347$	30.2586i	0.0872005i	0.999049	+	0.0436002i	$0.0138828\pi$
$347$	30.2586i	0.0872005i	−0.999049	+	0.0436002i	$0.986117\pi$
$348$	0	0
$349$	−440.357	−1.26177	−0.630884	−	0.775877i	$-0.717308\pi$
$349$	−440.357	−1.26177	−0.630884	+	0.775877i	$0.717308\pi$
$350$	72.8992i	0.208283i
$351$	0	0
$352$	266.886	0.758200
$353$	268.129i	0.759573i	0.925074	+	0.379787i	$0.124003\pi$
$353$	268.129i	0.759573i	−0.925074	+	0.379787i	$0.875997\pi$
$354$	0	0
$355$	313.752	0.883809
$356$	− 218.853i	− 0.614755i
$357$	0	0
$358$	−66.9581	−0.187034
$359$	− 226.280i	− 0.630307i	−0.949041	−	0.315153i	$-0.897944\pi$
$359$	− 226.280i	− 0.630307i	0.949041	−	0.315153i	$-0.102056\pi$
$360$	0	0
$361$	654.269	1.81238
$362$	108.474i	0.299652i
$363$	0	0
$364$	−203.767	−0.559801
$365$	− 533.276i	− 1.46103i
$366$	0	0
$367$	−546.177	−1.48822	−0.744110	−	0.668057i	$-0.767126\pi$
$367$	−546.177	−1.48822	−0.744110	+	0.668057i	$0.767126\pi$
$368$	135.372i	0.367860i
$369$	0	0
$370$	−154.447	−0.417423
$371$	190.206i	0.512685i
$372$	0	0
$373$	−105.024	−0.281566	−0.140783	−	0.990040i	$-0.544962\pi$
$373$	−105.024	−0.281566	−0.140783	+	0.990040i	$0.544962\pi$
$374$	− 182.324i	− 0.487496i
$375$	0	0
$376$	2.80137	0.00745044
$377$	149.232i	0.395842i
$378$	0	0
$379$	101.586	0.268037	0.134019	−	0.990979i	$-0.457212\pi$
$379$	101.586	0.268037	0.134019	+	0.990979i	$0.457212\pi$
$380$	− 880.437i	− 2.31694i
$381$	0	0
$382$	163.339	0.427589
$383$	− 233.716i	− 0.610223i	−0.952317	−	0.305112i	$-0.901306\pi$
$383$	− 233.716i	− 0.610223i	0.952317	−	0.305112i	$-0.0986938\pi$
$384$	0	0
$385$	194.487	0.505162
$386$	− 25.5959i	− 0.0663106i
$387$	0	0
$388$	−162.693	−0.419313
$389$	− 72.3016i	− 0.185865i	−0.995672	−	0.0929326i	$-0.970376\pi$
$389$	− 72.3016i	− 0.185865i	0.995672	−	0.0929326i	$-0.0296241\pi$
$390$	0	0
$391$	368.936	0.943571
$392$	− 37.8274i	− 0.0964985i
$393$	0	0
$394$	67.8473	0.172201
$395$	149.124i	0.377529i
$396$	0	0
$397$	−423.076	−1.06568	−0.532842	−	0.846215i	$-0.678876\pi$
$397$	−423.076	−1.06568	−0.532842	+	0.846215i	$0.678876\pi$
$398$	14.4560i	0.0363217i
$399$	0	0
$400$	−381.494	−0.953735
$401$	− 363.447i	− 0.906351i	−0.891421	−	0.453176i	$-0.850291\pi$
$401$	− 363.447i	− 0.906351i	0.891421	−	0.453176i	$-0.149709\pi$
$402$	0	0
$403$	−264.309	−0.655853
$404$	− 361.880i	− 0.895744i
$405$	0	0
$406$	−12.8843	−0.0317348
$407$	248.866i	0.611464i
$408$	0	0
$409$	−568.164	−1.38915	−0.694577	−	0.719418i	$-0.744408\pi$
$409$	−568.164	−1.38915	−0.694577	+	0.719418i	$0.744408\pi$
$410$	− 367.264i	− 0.895765i
$411$	0	0
$412$	−361.199	−0.876696
$413$	102.004i	0.246984i
$414$	0	0
$415$	636.331	1.53333
$416$	638.822i	1.53563i
$417$	0	0
$418$	213.040	0.509665
$419$	638.204i	1.52316i	0.648071	+	0.761580i	$0.275576\pi$
$419$	638.204i	1.52316i	−0.648071	+	0.761580i	$0.724424\pi$
$420$	0	0
$421$	246.608	0.585767	0.292883	−	0.956148i	$-0.405385\pi$
$421$	246.608	0.585767	0.292883	+	0.956148i	$0.405385\pi$
$422$	− 174.164i	− 0.412711i
$423$	0	0
$424$	388.494	0.916260
$425$	1039.70i	2.44636i
$426$	0	0
$427$	73.8753	0.173010
$428$	405.265i	0.946880i
$429$	0	0
$430$	93.1229	0.216565
$431$	138.307i	0.320898i	0.987044	+	0.160449i	$0.0512942\pi$
$431$	138.307i	0.320898i	−0.987044	+	0.160449i	$0.948706\pi$
$432$	0	0
$433$	209.792	0.484507	0.242254	−	0.970213i	$-0.422113\pi$
$433$	209.792	0.484507	0.242254	+	0.970213i	$0.422113\pi$
$434$	− 22.8198i	− 0.0525801i
$435$	0	0
$436$	−48.6320	−0.111541
$437$	431.091i	0.986479i
$438$	0	0
$439$	714.425	1.62739	0.813696	−	0.581291i	$-0.197452\pi$
$439$	714.425	1.62739	0.813696	+	0.581291i	$0.197452\pi$
$440$	− 397.238i	− 0.902813i
$441$	0	0
$442$	436.412	0.987357
$443$	− 523.302i	− 1.18127i	−0.806940	−	0.590634i	$-0.798878\pi$
$443$	− 523.302i	− 1.18127i	0.806940	−	0.590634i	$-0.201122\pi$
$444$	0	0
$445$	−499.990	−1.12357
$446$	259.196i	0.581157i
$447$	0	0
$448$	50.7370	0.113252
$449$	453.325i	1.00963i	0.863227	+	0.504816i	$0.168440\pi$
$449$	453.325i	1.00963i	−0.863227	+	0.504816i	$0.831560\pi$
$450$	0	0
$451$	−591.787	−1.31217
$452$	182.830i	0.404492i
$453$	0	0
$454$	−91.7195	−0.202025
$455$	465.527i	1.02314i
$456$	0	0
$457$	340.386	0.744826	0.372413	−	0.928067i	$-0.378530\pi$
$457$	340.386	0.744826	0.372413	+	0.928067i	$0.378530\pi$
$458$	173.442i	0.378694i
$459$	0	0
$460$	373.841	0.812697
$461$	− 198.359i	− 0.430280i	−0.976583	−	0.215140i	$-0.930979\pi$
$461$	− 198.359i	− 0.430280i	0.976583	−	0.215140i	$-0.0690208\pi$
$462$	0	0
$463$	578.341	1.24912	0.624559	−	0.780978i	$-0.285279\pi$
$463$	578.341	1.24912	0.624559	+	0.780978i	$0.285279\pi$
$464$	− 67.4260i	− 0.145315i
$465$	0	0
$466$	−255.585	−0.548467
$467$	867.984i	1.85864i	0.369277	+	0.929319i	$0.379605\pi$
$467$	867.984i	1.85864i	−0.369277	+	0.929319i	$0.620395\pi$
$468$	0	0
$469$	78.2212	0.166783
$470$	− 2.97651i	− 0.00633300i
$471$	0	0
$472$	208.343	0.441404
$473$	− 150.053i	− 0.317236i
$474$	0	0
$475$	−1214.86	−2.55760
$476$	− 250.911i	− 0.527125i
$477$	0	0
$478$	−333.140	−0.696946
$479$	− 723.856i	− 1.51118i	−0.655043	−	0.755591i	$-0.727350\pi$
$479$	− 723.856i	− 1.51118i	0.655043	−	0.755591i	$-0.272650\pi$
$480$	0	0
$481$	−595.688	−1.23844
$482$	206.885i	0.429223i
$483$	0	0
$484$	123.117	0.254375
$485$	371.689i	0.766369i
$486$	0	0
$487$	−881.932	−1.81095	−0.905474	−	0.424401i	$-0.860485\pi$
$487$	−881.932	−1.81095	−0.905474	+	0.424401i	$0.860485\pi$
$488$	− 150.889i	− 0.309200i
$489$	0	0
$490$	−40.1924	−0.0820253
$491$	751.995i	1.53156i	0.643103	+	0.765779i	$0.277647\pi$
$491$	751.995i	1.53156i	−0.643103	+	0.765779i	$0.722353\pi$
$492$	0	0
$493$	−183.759	−0.372736
$494$	509.935i	1.03226i
$495$	0	0
$496$	119.420	0.240766
$497$	104.479i	0.210218i
$498$	0	0
$499$	27.5359	0.0551822	0.0275911	−	0.999619i	$-0.491216\pi$
$499$	27.5359	0.0551822	0.0275911	+	0.999619i	$0.491216\pi$
$500$	362.730i	0.725460i
$501$	0	0
$502$	124.221	0.247453
$503$	− 820.917i	− 1.63204i	−0.578022	−	0.816021i	$-0.696175\pi$
$503$	− 820.917i	− 1.63204i	0.578022	−	0.816021i	$-0.303825\pi$
$504$	0	0
$505$	−826.751	−1.63713
$506$	90.4585i	0.178772i
$507$	0	0
$508$	80.4072	0.158282
$509$	490.271i	0.963203i	0.876390	+	0.481602i	$0.159945\pi$
$509$	490.271i	0.963203i	−0.876390	+	0.481602i	$0.840055\pi$
$510$	0	0
$511$	177.579	0.347513
$512$	− 504.914i	− 0.986160i
$513$	0	0
$514$	344.790	0.670798
$515$	825.194i	1.60232i
$516$	0	0
$517$	−4.79617	−0.00927693
$518$	− 51.4303i	− 0.0992862i
$519$	0	0
$520$	950.833	1.82852
$521$	424.710i	0.815182i	0.913164	+	0.407591i	$0.133631\pi$
$521$	424.710i	0.815182i	−0.913164	+	0.407591i	$0.866369\pi$
$522$	0	0
$523$	745.325	1.42510	0.712548	−	0.701624i	$-0.247541\pi$
$523$	745.325	1.42510	0.712548	+	0.701624i	$0.247541\pi$
$524$	− 181.896i	− 0.347129i
$525$	0	0
$526$	−223.826	−0.425524
$527$	− 325.460i	− 0.617570i
$528$	0	0
$529$	345.955	0.653979
$530$	− 412.783i	− 0.778836i
$531$	0	0
$532$	293.183	0.551096
$533$	− 1416.51i	− 2.65761i
$534$	0	0
$535$	925.867	1.73059
$536$	− 159.766i	− 0.298071i
$537$	0	0
$538$	1.69638	0.00315312
$539$	64.7637i	0.120155i
$540$	0	0
$541$	−2.11927	−0.00391732	−0.00195866	−	0.999998i	$-0.500623\pi$
$541$	−2.11927	−0.00391732	−0.00195866	+	0.999998i	$0.500623\pi$
$542$	− 9.46125i	− 0.0174562i
$543$	0	0
$544$	−786.621	−1.44599
$545$	111.105i	0.203862i
$546$	0	0
$547$	−172.013	−0.314466	−0.157233	−	0.987562i	$-0.550257\pi$
$547$	−172.013	−0.314466	−0.157233	+	0.987562i	$0.550257\pi$
$548$	− 505.289i	− 0.922059i
$549$	0	0
$550$	−254.922	−0.463494
$551$	− 214.717i	− 0.389686i
$552$	0	0
$553$	−49.6578	−0.0897971
$554$	− 161.331i	− 0.291211i
$555$	0	0
$556$	169.654	0.305133
$557$	866.241i	1.55519i	0.628765	+	0.777595i	$0.283561\pi$
$557$	866.241i	1.55519i	−0.628765	+	0.777595i	$0.716439\pi$
$558$	0	0
$559$	359.168	0.642518
$560$	− 210.334i	− 0.375596i
$561$	0	0
$562$	−30.8190	−0.0548380
$563$	494.890i	0.879023i	0.898237	+	0.439511i	$0.144848\pi$
$563$	494.890i	0.879023i	−0.898237	+	0.439511i	$0.855152\pi$
$564$	0	0
$565$	417.694	0.739281
$566$	74.9312i	0.132387i
$567$	0	0
$568$	213.396	0.375697
$569$	476.783i	0.837932i	0.908002	+	0.418966i	$0.137607\pi$
$569$	476.783i	0.837932i	−0.908002	+	0.418966i	$0.862393\pi$
$570$	0	0
$571$	724.636	1.26906	0.634532	−	0.772896i	$-0.281193\pi$
$571$	724.636	1.26906	0.634532	+	0.772896i	$0.281193\pi$
$572$	− 712.556i	− 1.24573i
$573$	0	0
$574$	122.298	0.213062
$575$	− 515.840i	− 0.897113i
$576$	0	0
$577$	74.4585	0.129044	0.0645221	−	0.997916i	$-0.479448\pi$
$577$	74.4585	0.129044	0.0645221	+	0.997916i	$0.479448\pi$
$578$	328.530i	0.568392i
$579$	0	0
$580$	−186.202	−0.321037
$581$	211.896i	0.364710i
$582$	0	0
$583$	−665.134	−1.14088
$584$	− 362.704i	− 0.621068i
$585$	0	0
$586$	−233.363	−0.398230
$587$	750.028i	1.27773i	0.769319	+	0.638865i	$0.220596\pi$
$587$	750.028i	1.27773i	−0.769319	+	0.638865i	$0.779404\pi$
$588$	0	0
$589$	380.290	0.645654
$590$	− 221.368i	− 0.375200i
$591$	0	0
$592$	269.143	0.454634
$593$	830.097i	1.39983i	0.714228	+	0.699913i	$0.246778\pi$
$593$	830.097i	1.39983i	−0.714228	+	0.699913i	$0.753222\pi$
$594$	0	0
$595$	−573.231	−0.963414
$596$	770.149i	1.29220i
$597$	0	0
$598$	−216.522	−0.362078
$599$	− 631.380i	− 1.05406i	−0.849848	−	0.527029i	$-0.823306\pi$
$599$	− 631.380i	− 1.05406i	0.849848	−	0.527029i	$-0.176694\pi$
$600$	0	0
$601$	545.316	0.907347	0.453674	−	0.891168i	$-0.350113\pi$
$601$	545.316	0.907347	0.453674	+	0.891168i	$0.350113\pi$
$602$	31.0096i	0.0515110i
$603$	0	0
$604$	−983.861	−1.62891
$605$	− 281.274i	− 0.464915i
$606$	0	0
$607$	5.26052	0.00866643	0.00433321	−	0.999991i	$-0.498621\pi$
$607$	5.26052	0.00866643	0.00433321	+	0.999991i	$0.498621\pi$
$608$	− 919.145i	− 1.51175i
$609$	0	0
$610$	−160.323	−0.262825
$611$	− 11.4802i	− 0.0187891i
$612$	0	0
$613$	−428.462	−0.698959	−0.349480	−	0.936944i	$-0.613642\pi$
$613$	−428.462	−0.698959	−0.349480	+	0.936944i	$0.613642\pi$
$614$	− 400.189i	− 0.651773i
$615$	0	0
$616$	132.279	0.214739
$617$	− 1045.71i	− 1.69483i	−0.530931	−	0.847415i	$-0.678158\pi$
$617$	− 1045.71i	− 1.69483i	0.530931	−	0.847415i	$-0.321842\pi$
$618$	0	0
$619$	−699.304	−1.12973	−0.564866	−	0.825182i	$-0.691072\pi$
$619$	−699.304	−1.12973	−0.564866	+	0.825182i	$0.691072\pi$
$620$	− 329.786i	− 0.531913i
$621$	0	0
$622$	203.706	0.327501
$623$	− 166.495i	− 0.267248i
$624$	0	0
$625$	−124.491	−0.199185
$626$	305.814i	0.488521i
$627$	0	0
$628$	119.629	0.190493
$629$	− 733.507i	− 1.16615i
$630$	0	0
$631$	−748.642	−1.18644	−0.593218	−	0.805042i	$-0.702143\pi$
$631$	−748.642	−1.18644	−0.593218	+	0.805042i	$0.702143\pi$
$632$	101.425i	0.160483i
$633$	0	0
$634$	117.812	0.185823
$635$	− 183.698i	− 0.289288i
$636$	0	0
$637$	−155.019	−0.243358
$638$	− 45.0554i	− 0.0706197i
$639$	0	0
$640$	−1026.88	−1.60450
$641$	66.0855i	0.103097i	0.998670	+	0.0515487i	$0.0164158\pi$
$641$	66.0855i	0.103097i	−0.998670	+	0.0515487i	$0.983584\pi$
$642$	0	0
$643$	763.206	1.18695	0.593473	−	0.804854i	$-0.297757\pi$
$643$	763.206	1.18695	0.593473	+	0.804854i	$0.297757\pi$
$644$	124.488i	0.193304i
$645$	0	0
$646$	−627.914	−0.972004
$647$	− 522.507i	− 0.807585i	−0.914851	−	0.403792i	$-0.867692\pi$
$647$	− 522.507i	− 0.807585i	0.914851	−	0.403792i	$-0.132308\pi$
$648$	0	0
$649$	−356.700	−0.549614
$650$	− 610.183i	− 0.938744i
$651$	0	0
$652$	−1.78871	−0.00274342
$653$	38.5504i	0.0590359i	0.999564	+	0.0295179i	$0.00939721\pi$
$653$	38.5504i	0.0590359i	−0.999564	+	0.0295179i	$0.990603\pi$
$654$	0	0
$655$	−415.558	−0.634440
$656$	640.005i	0.975618i
$657$	0	0
$658$	0.991169	0.00150634
$659$	749.071i	1.13668i	0.822794	+	0.568339i	$0.192414\pi$
$659$	749.071i	1.13668i	−0.822794	+	0.568339i	$0.807586\pi$
$660$	0	0
$661$	605.366	0.915834	0.457917	−	0.888995i	$-0.348596\pi$
$661$	605.366	0.915834	0.457917	+	0.888995i	$0.348596\pi$
$662$	− 4.11214i	− 0.00621169i
$663$	0	0
$664$	432.796	0.651801
$665$	− 669.805i	− 1.00723i
$666$	0	0
$667$	91.1706	0.136688
$668$	− 543.194i	− 0.813164i
$669$	0	0
$670$	−169.755	−0.253365
$671$	258.335i	0.385000i
$672$	0	0
$673$	1042.10	1.54844	0.774222	−	0.632914i	$-0.218141\pi$
$673$	1042.10	1.54844	0.774222	+	0.632914i	$0.218141\pi$
$674$	464.151i	0.688652i
$675$	0	0
$676$	1117.84	1.65361
$677$	42.8646i	0.0633155i	0.999499	+	0.0316577i	$0.0100787\pi$
$677$	42.8646i	0.0633155i	−0.999499	+	0.0316577i	$0.989921\pi$
$678$	0	0
$679$	−123.771	−0.182285
$680$	1170.82i	1.72179i
$681$	0	0
$682$	79.7986	0.117007
$683$	− 1041.21i	− 1.52446i	−0.647305	−	0.762231i	$-0.724104\pi$
$683$	− 1041.21i	− 1.52446i	0.647305	−	0.762231i	$-0.275896\pi$
$684$	0	0
$685$	−1154.38	−1.68523
$686$	− 13.3840i	− 0.0195101i
$687$	0	0
$688$	−162.279	−0.235870
$689$	− 1592.07i	− 2.31070i
$690$	0	0
$691$	−521.023	−0.754014	−0.377007	−	0.926210i	$-0.623047\pi$
$691$	−521.023	−0.754014	−0.377007	+	0.926210i	$0.623047\pi$
$692$	− 265.603i	− 0.383819i
$693$	0	0
$694$	21.8668	0.0315084
$695$	− 387.591i	− 0.557685i
$696$	0	0
$697$	1744.23	2.50249
$698$	318.231i	0.455918i
$699$	0	0
$700$	−350.820	−0.501171
$701$	− 249.669i	− 0.356161i	−0.984016	−	0.178080i	$-0.943011\pi$
$701$	− 249.669i	− 0.356161i	0.984016	−	0.178080i	$-0.0569887\pi$
$702$	0	0
$703$	857.083	1.21918
$704$	177.423i	0.252021i
$705$	0	0
$706$	193.768	0.274459
$707$	− 275.306i	− 0.389400i
$708$	0	0
$709$	1050.17	1.48120	0.740601	−	0.671945i	$-0.234541\pi$
$709$	1050.17	1.48120	0.740601	+	0.671945i	$0.234541\pi$
$710$	− 226.738i	− 0.319349i
$711$	0	0
$712$	−340.065	−0.477619
$713$	161.474i	0.226472i
$714$	0	0
$715$	−1627.90	−2.27679
$716$	− 322.229i	− 0.450041i
$717$	0	0
$718$	−163.525	−0.227750
$719$	− 121.143i	− 0.168488i	−0.996445	−	0.0842441i	$-0.973152\pi$
$719$	− 121.143i	− 0.168488i	0.996445	−	0.0842441i	$-0.0268476\pi$
$720$	0	0
$721$	−274.787	−0.381120
$722$	− 472.818i	− 0.654873i
$723$	0	0
$724$	−522.020	−0.721022
$725$	256.928i	0.354384i
$726$	0	0
$727$	961.438	1.32247	0.661236	−	0.750178i	$-0.270032\pi$
$727$	961.438	1.32247	0.661236	+	0.750178i	$0.270032\pi$
$728$	316.624i	0.434924i
$729$	0	0
$730$	−385.380	−0.527918
$731$	442.265i	0.605014i
$732$	0	0
$733$	161.183	0.219895	0.109948	−	0.993937i	$-0.464932\pi$
$733$	161.183	0.219895	0.109948	+	0.993937i	$0.464932\pi$
$734$	394.703i	0.537743i
$735$	0	0
$736$	390.276	0.530267
$737$	273.533i	0.371143i
$738$	0	0
$739$	−643.023	−0.870126	−0.435063	−	0.900400i	$-0.643274\pi$
$739$	−643.023	−0.870126	−0.435063	+	0.900400i	$0.643274\pi$
$740$	− 743.258i	− 1.00440i
$741$	0	0
$742$	137.456	0.185250
$743$	49.7283i	0.0669291i	0.999440	+	0.0334646i	$0.0106541\pi$
$743$	49.7283i	0.0669291i	−0.999440	+	0.0334646i	$0.989346\pi$
$744$	0	0
$745$	1759.48	2.36172
$746$	75.8973i	0.101739i
$747$	0	0
$748$	877.414	1.17301
$749$	308.311i	0.411630i
$750$	0	0
$751$	966.138	1.28647	0.643235	−	0.765669i	$-0.277592\pi$
$751$	966.138	1.28647	0.643235	+	0.765669i	$0.277592\pi$
$752$	5.18696i	0.00689755i
$753$	0	0
$754$	107.845	0.143031
$755$	2247.73i	2.97712i
$756$	0	0
$757$	136.110	0.179802	0.0899009	−	0.995951i	$-0.471345\pi$
$757$	136.110	0.179802	0.0899009	+	0.995951i	$0.471345\pi$
$758$	− 73.4127i	− 0.0968506i
$759$	0	0
$760$	−1368.07	−1.80009
$761$	− 1340.82i	− 1.76191i	−0.473196	−	0.880957i	$-0.656900\pi$
$761$	− 1340.82i	− 1.76191i	0.473196	−	0.880957i	$-0.343100\pi$
$762$	0	0
$763$	−36.9975	−0.0484895
$764$	786.052i	1.02886i
$765$	0	0
$766$	−168.898	−0.220494
$767$	− 853.800i	− 1.11317i
$768$	0	0
$769$	−572.189	−0.744069	−0.372035	−	0.928219i	$-0.621340\pi$
$769$	−572.189	−0.744069	−0.372035	+	0.928219i	$0.621340\pi$
$770$	− 140.549i	− 0.182531i
$771$	0	0
$772$	123.178	0.159556
$773$	628.925i	0.813616i	0.913514	+	0.406808i	$0.133358\pi$
$773$	628.925i	0.813616i	−0.913514	+	0.406808i	$0.866642\pi$
$774$	0	0
$775$	−455.052	−0.587164
$776$	252.801i	0.325775i
$777$	0	0
$778$	−52.2499	−0.0671592
$779$	2038.09i	2.61629i
$780$	0	0
$781$	−365.352	−0.467800
$782$	− 266.617i	− 0.340943i
$783$	0	0
$784$	70.0405	0.0893374
$785$	− 273.305i	− 0.348159i
$786$	0	0
$787$	−266.123	−0.338149	−0.169074	−	0.985603i	$-0.554078\pi$
$787$	−266.123	−0.338149	−0.169074	+	0.985603i	$0.554078\pi$
$788$	326.508i	0.414350i
$789$	0	0
$790$	107.767	0.136414
$791$	139.091i	0.175842i
$792$	0	0
$793$	−618.354	−0.779765
$794$	305.743i	0.385066i
$795$	0	0
$796$	−69.5681	−0.0873971
$797$	716.744i	0.899303i	0.893204	+	0.449651i	$0.148452\pi$
$797$	716.744i	0.899303i	−0.893204	+	0.449651i	$0.851548\pi$
$798$	0	0
$799$	14.1362	0.0176924
$800$	1099.84i	1.37480i
$801$	0	0
$802$	−262.651	−0.327494
$803$	620.979i	0.773323i
$804$	0	0
$805$	284.405	0.353298
$806$	191.007i	0.236981i
$807$	0	0
$808$	−562.309	−0.695927
$809$	− 591.782i	− 0.731498i	−0.930713	−	0.365749i	$-0.880813\pi$
$809$	− 591.782i	− 0.731498i	0.930713	−	0.365749i	$-0.119187\pi$
$810$	0	0
$811$	−1308.23	−1.61311	−0.806555	−	0.591159i	$-0.798670\pi$
$811$	−1308.23	−1.61311	−0.806555	+	0.591159i	$0.798670\pi$
$812$	− 62.0046i	− 0.0763603i
$813$	0	0
$814$	179.847	0.220942
$815$	4.08648i	0.00501408i
$816$	0	0
$817$	−516.774	−0.632527
$818$	410.593i	0.501947i
$819$	0	0
$820$	1767.42	2.15539
$821$	142.567i	0.173650i	0.996224	+	0.0868252i	$0.0276722\pi$
$821$	142.567i	0.173650i	−0.996224	+	0.0868252i	$0.972328\pi$
$822$	0	0
$823$	113.006	0.137310	0.0686550	−	0.997640i	$-0.478129\pi$
$823$	113.006	0.137310	0.0686550	+	0.997640i	$0.478129\pi$
$824$	561.250i	0.681128i
$825$	0	0
$826$	73.7150	0.0892433
$827$	69.0940i	0.0835477i	0.999127	+	0.0417739i	$0.0133009\pi$
$827$	69.0940i	0.0835477i	−0.999127	+	0.0417739i	$0.986699\pi$
$828$	0	0
$829$	−107.675	−0.129885	−0.0649426	−	0.997889i	$-0.520686\pi$
$829$	−107.675	−0.129885	−0.0649426	+	0.997889i	$0.520686\pi$
$830$	− 459.855i	− 0.554042i
$831$	0	0
$832$	−424.681	−0.510434
$833$	− 190.884i	− 0.229153i
$834$	0	0
$835$	−1240.98	−1.48620
$836$	1025.23i	1.22636i
$837$	0	0
$838$	461.208	0.550368
$839$	462.914i	0.551745i	0.961194	+	0.275873i	$0.0889668\pi$
$839$	462.914i	0.551745i	−0.961194	+	0.275873i	$0.911033\pi$
$840$	0	0
$841$	795.590	0.946005
$842$	− 178.215i	− 0.211657i
$843$	0	0
$844$	838.147	0.993065
$845$	− 2553.82i	− 3.02227i
$846$	0	0
$847$	93.6633	0.110582
$848$	719.329i	0.848265i
$849$	0	0
$850$	751.356	0.883949
$851$	363.924i	0.427643i
$852$	0	0
$853$	−890.954	−1.04449	−0.522247	−	0.852794i	$-0.674906\pi$
$853$	−890.954	−1.04449	−0.522247	+	0.852794i	$0.674906\pi$
$854$	− 53.3871i	− 0.0625142i
$855$	0	0
$856$	629.722	0.735656
$857$	− 615.003i	− 0.717623i	−0.933410	−	0.358811i	$-0.883182\pi$
$857$	− 615.003i	− 0.717623i	0.933410	−	0.358811i	$-0.116818\pi$
$858$	0	0
$859$	34.4978	0.0401604	0.0200802	−	0.999798i	$-0.493608\pi$
$859$	34.4978	0.0401604	0.0200802	+	0.999798i	$0.493608\pi$
$860$	448.144i	0.521098i
$861$	0	0
$862$	99.9496	0.115951
$863$	588.277i	0.681665i	0.940124	+	0.340832i	$0.110709\pi$
$863$	588.277i	0.681665i	−0.940124	+	0.340832i	$0.889291\pi$
$864$	0	0
$865$	−606.796	−0.701498
$866$	− 151.609i	− 0.175068i
$867$	0	0
$868$	109.818	0.126518
$869$	− 173.649i	− 0.199826i
$870$	0	0
$871$	−654.730	−0.751700
$872$	75.5669i	0.0866593i
$873$	0	0
$874$	311.535	0.356447
$875$	275.952i	0.315374i
$876$	0	0
$877$	66.4285	0.0757452	0.0378726	−	0.999283i	$-0.487942\pi$
$877$	66.4285	0.0757452	0.0378726	+	0.999283i	$0.487942\pi$
$878$	− 516.290i	− 0.588030i
$879$	0	0
$880$	735.518	0.835816
$881$	− 1060.27i	− 1.20349i	−0.798689	−	0.601744i	$-0.794473\pi$
$881$	− 1060.27i	− 1.20349i	0.798689	−	0.601744i	$-0.205527\pi$
$882$	0	0
$883$	828.431	0.938201	0.469100	−	0.883145i	$-0.344578\pi$
$883$	828.431	0.938201	0.469100	+	0.883145i	$0.344578\pi$
$884$	2100.19i	2.37578i
$885$	0	0
$886$	−378.172	−0.426831
$887$	285.577i	0.321958i	0.986958	+	0.160979i	$0.0514651\pi$
$887$	285.577i	0.321958i	−0.986958	+	0.160979i	$0.948535\pi$
$888$	0	0
$889$	61.1709	0.0688087
$890$	361.326i	0.405984i
$891$	0	0
$892$	−1247.35	−1.39838
$893$	16.5178i	0.0184970i
$894$	0	0
$895$	−736.164	−0.822530
$896$	− 341.949i	− 0.381639i
$897$	0	0
$898$	327.603	0.364814
$899$	− 80.4268i	− 0.0894625i
$900$	0	0
$901$	1960.42	2.17582
$902$	427.664i	0.474129i
$903$	0	0
$904$	284.091	0.314260
$905$	1192.61i	1.31780i
$906$	0	0
$907$	766.603	0.845207	0.422604	−	0.906315i	$-0.361116\pi$
$907$	766.603	0.845207	0.422604	+	0.906315i	$0.361116\pi$
$908$	− 441.391i	− 0.486113i
$909$	0	0
$910$	336.420	0.369692
$911$	1099.95i	1.20741i	0.797207	+	0.603706i	$0.206310\pi$
$911$	1099.95i	1.20741i	−0.797207	+	0.603706i	$0.793690\pi$
$912$	0	0
$913$	−740.982	−0.811591
$914$	− 245.985i	− 0.269130i
$915$	0	0
$916$	−834.672	−0.911214
$917$	− 138.380i	− 0.150905i
$918$	0	0
$919$	411.489	0.447757	0.223879	−	0.974617i	$-0.428128\pi$
$919$	411.489	0.447757	0.223879	+	0.974617i	$0.428128\pi$
$920$	− 580.893i	− 0.631405i
$921$	0	0
$922$	−143.347	−0.155474
$923$	− 874.510i	− 0.947465i
$924$	0	0
$925$	−1025.58	−1.10873
$926$	− 417.947i	− 0.451347i
$927$	0	0
$928$	−194.388	−0.209470
$929$	217.553i	0.234180i	0.993121	+	0.117090i	$0.0373566\pi$
$929$	217.553i	0.234180i	−0.993121	+	0.117090i	$0.962643\pi$
$930$	0	0
$931$	223.043	0.239574
$932$	− 1229.98i	− 1.31972i
$933$	0	0
$934$	627.262	0.671587
$935$	− 2004.54i	− 2.14389i
$936$	0	0
$937$	−674.025	−0.719344	−0.359672	−	0.933079i	$-0.617111\pi$
$937$	−674.025	−0.719344	−0.359672	+	0.933079i	$0.617111\pi$
$938$	− 56.5278i	− 0.0602642i
$939$	0	0
$940$	14.3242	0.0152385
$941$	− 1535.74i	− 1.63203i	−0.578030	−	0.816015i	$-0.696178\pi$
$941$	− 1535.74i	− 1.63203i	0.578030	−	0.816015i	$-0.303822\pi$
$942$	0	0
$943$	−865.388	−0.917697
$944$	385.763i	0.408647i
$945$	0	0
$946$	−108.438	−0.114628
$947$	− 312.347i	− 0.329828i	−0.986308	−	0.164914i	$-0.947265\pi$
$947$	− 312.347i	− 0.329828i	0.986308	−	0.164914i	$-0.0527346\pi$
$948$	0	0
$949$	−1486.38	−1.56626
$950$	877.939i	0.924146i
$951$	0	0
$952$	−389.879	−0.409537
$953$	842.068i	0.883597i	0.897114	+	0.441798i	$0.145659\pi$
$953$	842.068i	0.883597i	−0.897114	+	0.441798i	$0.854341\pi$
$954$	0	0
$955$	1795.81	1.88043
$956$	− 1603.20i	− 1.67699i
$957$	0	0
$958$	−523.106	−0.546040
$959$	− 384.406i	− 0.400840i
$960$	0	0
$961$	−818.554	−0.851773
$962$	430.483i	0.447488i
$963$	0	0
$964$	−995.615	−1.03280
$965$	− 281.411i	− 0.291618i
$966$	0	0
$967$	1270.48	1.31383	0.656916	−	0.753964i	$-0.271861\pi$
$967$	1270.48	1.31383	0.656916	+	0.753964i	$0.271861\pi$
$968$	− 191.306i	− 0.197630i
$969$	0	0
$970$	268.607	0.276914
$971$	− 140.610i	− 0.144810i	−0.997375	−	0.0724048i	$-0.976933\pi$
$971$	− 140.610i	− 0.144810i	0.997375	−	0.0724048i	$-0.0230673\pi$
$972$	0	0
$973$	129.067	0.132648
$974$	637.342i	0.654355i
$975$	0	0
$976$	279.384	0.286254
$977$	824.093i	0.843493i	0.906714	+	0.421747i	$0.138583\pi$
$977$	824.093i	0.843493i	−0.906714	+	0.421747i	$0.861417\pi$
$978$	0	0
$979$	582.219	0.594708
$980$	− 193.422i	− 0.197369i
$981$	0	0
$982$	543.441	0.553403
$983$	191.958i	0.195277i	0.995222	+	0.0976387i	$0.0311290\pi$
$983$	191.958i	0.195277i	−0.995222	+	0.0976387i	$0.968871\pi$
$984$	0	0
$985$	745.940	0.757299
$986$	132.796i	0.134682i
$987$	0	0
$988$	−2454.01	−2.48382
$989$	− 219.426i	− 0.221867i
$990$	0	0
$991$	1913.96	1.93135	0.965673	−	0.259761i	$-0.0836438\pi$
$991$	1913.96	1.93135	0.965673	+	0.259761i	$0.0836438\pi$
$992$	− 344.285i	− 0.347061i
$993$	0	0
$994$	75.5030	0.0759588
$995$	158.935i	0.159734i
$996$	0	0
$997$	−1686.26	−1.69134	−0.845669	−	0.533708i	$-0.820798\pi$
$997$	−1686.26	−1.69134	−0.845669	+	0.533708i	$0.820798\pi$
$998$	− 19.8993i	− 0.0199391i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.b.c.134.4	✓	8
3.2	odd	2	inner	189.3.b.c.134.5	yes	8
4.3	odd	2		3024.3.d.j.1457.1		8
9.2	odd	6		567.3.r.e.134.5		16
9.4	even	3		567.3.r.e.512.5		16
9.5	odd	6		567.3.r.e.512.4		16
9.7	even	3		567.3.r.e.134.4		16
12.11	even	2		3024.3.d.j.1457.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.b.c.134.4	✓	8	1.1	even	1	trivial
189.3.b.c.134.5	yes	8	3.2	odd	2	inner
567.3.r.e.134.4		16	9.7	even	3
567.3.r.e.134.5		16	9.2	odd	6
567.3.r.e.512.4		16	9.5	odd	6
567.3.r.e.512.5		16	9.4	even	3
3024.3.d.j.1457.1		8	4.3	odd	2
3024.3.d.j.1457.8		8	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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	189.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.722666i q^{2} +3.47775 q^{4} -7.94527i q^{5} +2.64575 q^{7} -5.40392i q^{8} +O(q^{10})\)	\(q-0.722666i q^{2} +3.47775 q^{4} -7.94527i q^{5} +2.64575 q^{7} -5.40392i q^{8} -5.74177 q^{10} +9.25195i q^{11} -22.1456 q^{13} -1.91199i q^{14} +10.0058 q^{16} -27.2692i q^{17} +31.8633 q^{19} -27.6317i q^{20} +6.68607 q^{22} +13.5294i q^{23} -38.1273 q^{25} +16.0038i q^{26} +9.20127 q^{28} -6.73870i q^{29} +11.9351 q^{31} -28.8465i q^{32} -19.7065 q^{34} -21.0212i q^{35} +26.8988 q^{37} -23.0265i q^{38} -42.9356 q^{40} +63.9635i q^{41} -16.2185 q^{43} +32.1760i q^{44} +9.77724 q^{46} +0.518396i q^{47} +7.00000 q^{49} +27.5533i q^{50} -77.0168 q^{52} +71.8912i q^{53} +73.5092 q^{55} -14.2974i q^{56} -4.86983 q^{58} +38.5540i q^{59} +27.9222 q^{61} -8.62506i q^{62} +19.1768 q^{64} +175.953i q^{65} +29.5649 q^{67} -94.8356i q^{68} -15.1913 q^{70} +39.4892i q^{71} +67.1187 q^{73} -19.4388i q^{74} +110.813 q^{76} +24.4784i q^{77} -18.7689 q^{79} -79.4987i q^{80} +46.2242 q^{82} +80.0893i q^{83} -216.661 q^{85} +11.7205i q^{86} +49.9968 q^{88} -62.9293i q^{89} -58.5917 q^{91} +47.0520i q^{92} +0.374627 q^{94} -253.162i q^{95} -46.7811 q^{97} -5.05866i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{4}+O(q^{10}) \) 8 * q - 12 * q^4	\( 8 q - 12 q^{4} - 52 q^{10} + 36 q^{13} + 132 q^{16} + 12 q^{19} - 136 q^{22} - 108 q^{25} + 56 q^{28} - 28 q^{31} - 12 q^{34} - 4 q^{37} + 336 q^{40} - 152 q^{43} + 108 q^{46} + 56 q^{49} - 272 q^{52} + 196 q^{55} - 220 q^{58} + 180 q^{61} - 700 q^{64} - 132 q^{67} + 196 q^{70} + 272 q^{73} + 544 q^{76} + 316 q^{79} + 28 q^{82} - 228 q^{85} - 56 q^{91} - 348 q^{94} - 364 q^{97}+O(q^{100}) \) 8 * q - 12 * q^4 - 52 * q^10 + 36 * q^13 + 132 * q^16 + 12 * q^19 - 136 * q^22 - 108 * q^25 + 56 * q^28 - 28 * q^31 - 12 * q^34 - 4 * q^37 + 336 * q^40 - 152 * q^43 + 108 * q^46 + 56 * q^49 - 272 * q^52 + 196 * q^55 - 220 * q^58 + 180 * q^61 - 700 * q^64 - 132 * q^67 + 196 * q^70 + 272 * q^73 + 544 * q^76 + 316 * q^79 + 28 * q^82 - 228 * q^85 - 56 * q^91 - 348 * q^94 - 364 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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